NANO LETTERS 2005 Imperfect Perfect Lens Vol. 5, No. 2 - Ivan A. Larkin and Mark I. Stockman* 339 343 Department of Physics and Astronomy, Georgia State UniVersity, Atlanta, Georgia 30303 Received December 9, 2004; Revised Manuscript Received January 12, 2005 ABSTRACT We have quantitatively established a fundamental limitation on the ultimate spatial resolution of the perfect lens (thin metal slab) in the near field. This limitation stems from the spatial dispersion of the dielectric response of the Fermi liquid of electrons with Coulomb interaction in the metal. We discuss possible applications in nanoimaging, nanophotolithography, and nanospectroscopy. The Veselago lens,1 that is a slab of a left-handed material We consider only near-field imaging and therefore use the with relative dielectric permittivity )-1 and magnetic quasistatic approximation, which does not include magnetic permeability µ )-1, builds a perfect image of a three- field. Therefore, the left-handedness of the material is dimensional object in the far-field zone. Such a lens was irrelevant. Consider a metal ( < 0) slab of thickness b experimentally confirmed in the microwave spectral region.2 parallel to the xy plane of the coordinate system, whose As shown by Pendry, the same slab builds also a perfect boundaries are at z ) 0 and z ) b. Let there be a unit charge three-dimensional image in the near field, deserving the name at a point (0, 0, - a). Then the electric potential to the right of perfect lens,3 as observed later in ref 4. In the near zone, of the slab (z > b) is given by8 the requirement for the permeability is lifted, and only the )- condition of 1 is important. -kr(z+a-2b) ∞ J (k r)e Actually, such a lens is only near-perfect,3 because its æ(r, z) ) 4∫ 0 r dk (1) 0 2 2k b 2 r spatial resolution is limited by the losses in the lens’s (1 + ) e r - (1 - ) material; in the visible range the best material is silver, having the lowest losses. One of the points of this letter is to suggest where is the metal permittivity relative to the host medium, that such losses can be minimized by using a high-index x 2 2 J0 is a Bessel function, r ) x +y is the radial coordinate, embedding dielectric that would shift the operational fre- and kr is a two-dimensional (2D) wave vector in the plane quency toward the red where the quality factor of silver is of the slab. For )-1 (i.e., for the frequency of the surface 5 much higher. Light amplification can also be used to plasmon resonance of the metal), behind the image plane of 6 compensate these losses. z g 2b - a, this potential is an ideal three-dimensional (3D) In this letter, we investigate a principal limitation on the image of the original single point charge resolution of the perfect lens stemming from the spatial dispersion (nonlocality) of the dielectric response of the ∞ - + - æ(r, z) ) ∫ J (k r)e kr(z a 2b) dk Fermi liquid of interacting electrons in the nanolens material. 0 0 r r Such dispersion leads to the aberrations of this lens in the 1 wave vector in the plane of the slab. This principally limits ) (2) the resolution of this lens, making it imperfect on the scale xr2 + (z - 2b + a)2 below ∼5 nanometers. This effect is independent from the known sources of the lens imperfection due to optical losses If is not exactly equal to -1, e.g., due to Im * 0, then in the metal and temporal dispersion that is related to these the image of eq 1 is not ideal, limited by the spatial resolution losses by Kramers-Kronig relations. Even if the absorption of δ ∼ b/(- ln| + 1|) g b/ln Q, where Q )-Re/Im is is reduced or compensated by the optical gain as in ref 6, the quality factor of the plasmon resonance of the metal. the spatial dispersion will remain and limit the resolution. The highest value of Q ∼ 100 for any natural material from The principal role of spatial dispersion of the surface polar- the near-infrared to ultraviolet region is for silver for light itons at high wavenumbers was pointed out two years ago frequency pω ≈ 1.2 eV. The surface plasmon resonance in preprint7 where it was considered for left-handed ( < 0 would have shifted to this frequency if the values of host and µ < 0) systems. Only a generic electromagnetic spectrum dielectric constant h ∼ 100 were available. However, it is was considered, and no quantitative limitations on the spatial not the case, and the realistic values of h ≈ 12 are typical 9 resolution were obtained, in contrast to the present letter. for wide-band semiconductors (e.g., Al0.2Ga0.8As) for pω 10.1021/nl047957a CCC: $30.25 © 2005 American Chemical Society Published on Web 01/22/2005 L R ) 2.2 eV. This choice provides the best possible suppression where E (kr) and E (kr) are the 2D Fourier transforms of L R of the absorption adverse effects. E⊥(x, y,0)andE⊥(x, y, b). Deriving eq 4, we have taken Now we consider a principally different source of the lens into account the cylindrical symmetry of the problem. This imperfection, namely spatial dispersion,ornonlocality of potential should be found consistently with the potentials the dielectric response function (ω,k) that depends not only outside of the metal slab, where they obey the Laplace on frequency ω but also on wave vector k. In metals, the equation whose solutions to the left (L) and right (R) of the characteristic wave-vector scale of (ω, k)isk ∼ w/VF, where slab are VF is the electron velocity at the Fermi surface. For k g |ω|/ V , there exists the Landau damping that additionally L )RL + L - F æ (kr, z) (kr) exp(krz) â (kr) exp( krz) contributes to the optical losses described by Im(ω, k).10 R )RR - + L - - For such a spatially dispersive dielectric function, we æ (kr, z) (kr) exp[kr(z b)] â (kr) exp[ kr(z b)] should carefully reformulate the boundary conditions for (6) ballistic electrons at the surface of the metal. We will consider density of electrons in the metal as constant From eq 5, performing the inverse Fourier transform over neglecting the narrow (width on order of the Debye screening kz, we express the continuity equation for the left face of radius rD ∼ 0.1 nm) Schottky layer in the metal at the the metal slab as interface. We assume that the semiconductor of the embed- ding medium is undoped and neglect Schottky-layer electric RL + L ) 1 L -V R (kr) â (kr) [uE (kr) E (kr)] (7) fields inside the semiconductor, because they are too weak kr to change its dielectric function. We also neglect the Friedel 10 ∼ ∼ oscillations because their period is too short, 1/kF 0.1 where nm, where kF is the electron wave vector at the Fermi surface. To take into account the effect of boundaries, we assume ∞ kr ∞ exp(2ikznb) specular reflection of electrons from the surface. As was ) u ∑ ∫-∞ dkz 2 shown in ref 11, the ideal specular reflection implies that a πn)-∞ k (ω, k) nonlocal electrostatic problem for a half-infinite metal is ∞ - kr exp[ikz(2n 1)b] equivalent to a problem in bulk metal V) ∞ ∑ ∫-∞ dkz (8) 2 πn)-∞ k (ω, k) ∇ ∇ ) (ˆ æi) 2E⊥(x, y,0)δ(z) (3) Similarly for z ) b (the right face of the slab), we get where E⊥(x, y, 0) is the normal electric field at this surface, R R 1 L R æi is the field potential in the metal, and ˆ is the nonlocal R + ) V - (kr) â (kr) [ E (kr) uE (kr)] (9) dielectric response operator. The Dirac delta function emu- kr lates the boundary condition at the metal surface. For the slab, an electron experiences multiple reflections The other two equations come from the displacement from both its surfaces. To correspondingly generalize eq 3, matching, we reflect each boundary periodically in each other. As a result, the potential æ satisfies an equation L ) RL - L i E (kr) Lkr[ (kr) â (kr)] R ) RR - R ∞ E (kr) Rkr[ (kr) â (kr)] (10) L ∇(ˆ∇æ ) ) 2 ∑ {E⊥(x, y,0)δ(z - 2bn) - i n)-∞ where L and R are the host dielectric permittivities to the 1 R - + left and right of the lens (metal slab). E⊥(x, y, b) δ[z 2b(n )]} (4) 2 Solving eqs 7, 9, and 10 for RR and âR,weget where EL(x, y,0)andER(x, y, b) are the electric fields at the RR RL ⊥ ⊥ ( ) ) Tˆ·( ) (11) left and right faces of the slab. âR âL The solution of eq 4 yields a periodic potential of period 2b in the z direction with discontinuities of the normal electric where Tˆ isa2× 2 transfer matrix with matrix elements L R field 2E⊥(x, y,0)atz ) 2nb and 2E⊥(x, y, b)atz ) (2n + ) V2 - 2 + + - V 1)b.Inthek space T11 [( u )RL u(R L) 1]/(2 R) T ) [(u2 -V2) + u( - ) - 1]/(2V ) 2 ∞ 12 R L R L R ) { L - æ(k) ∑ E (kr) exp(2ikznb) 2 2 2 T ) [(V - u ) + u( - ) + 1]/(2V ) k (ω, k) n)-∞ 21 R L R L R R + } ) 2 -V2 + + + V E (kr) exp[i(2n 1)kzb] (5) T22 [(u )RL u(R L) 1]/(2 R) (12) 340 Nano Lett., Vol.