April 1, 2009 / Vol. 34, No. 7 / OPTICS LETTERS 1015

Perfect lens with not so perfect boundaries

P. C. Ingrey,* K. I. Hopcraft, O. French, and E. JakemanSchool of Mathematical Sciences, University of Nottingham, University Park, NG7 2RD, England

*Corresponding author: pmxpi@nottingham.ac.uk

Received January 13, 2009; revised February 10, 2009; accepted February 18, 2009;posted February 24, 2009 (Doc. ID 106316); published March 24, 2009

In manufacturing left-handed media the interfaces will never be perfect; defects and other disturbances tointerfaces and material parameters are unavoidable. We report an analytical calculation of electromagneticwave propagation through a perfect lens with diffuse boundaries. Field localizations are generated in theboundary layers, and the lens’ ability to recover evanescent modes in the presence of these boundaries isanalyzed and quantified. It is shown that such a diffuse layer produces an effect that is qualitatively similarto a lens with increased losses. © 2009 Optical Society of America

OCIS codes: 080.3630, 160.3918, 240.5770, 350.3618.

Recently materials have been constructed with a re-fractive index defined by microscopic configurationsof manmade elements rather than atomic structures.Materials fabricated in this manner, called metama-terials, have generated a surge of research interestsince they can be tailored to have properties that donot occur naturally [1–4]. Prime examples of this areleft-handed media (LHM), which simultaneouslyhave negative permeability and permittivity over aband of frequencies. Causal arguments then implythe refractive index must have a negative real part[5]. Furthermore, a slab of LHM with n=−1 can forma “perfect lens” due to its ability to focus both propa-gating and evanescent modes [6], leading to en-hanced resolution beyond the conventional diffrac-tion limit.

Factors affecting the performance of LHM struc-tures are numerous [7–10]. However, the principal ef-fects are losses within the media; the frequencies forwhich the media are left-handed and surface effects.The first two of these are now well understood, andtherefore we turn our attention to the third: defects,perturbations, or foreign bodies on the surface.

Recently a graded-index (GRIN) model was used toinvestigate surface effects [11]. GRIN modeling ofsurface roughness, where the rough surface is re-placed by a smooth transition through a layer be-tween bulk media, is a well known technique fortreating conventional right-handed materials(RHMs) [12].

This Letter is principally concerned with the per-fect lens’ ability to resolve evanescent modes in thepresence of GRIN, or “diffuse,” boundaries. The su-perresolving ability of the lens is governed by evanes-cent waves, and their behavior is quantified as afunction of the properties of the GRIN layer and thebulk LHM. In particular it will be shown that there isa qualitative similarity between the effects producedby diffuse boundaries and those caused by loss in thebulk medium.

We model the lens as a slab of LHM of refractiveindex n=−1+�i with a diffuse layer on both inter-faces with air in which n=1. The permeability andpermittivity change linearly through the layer ac-cording to ��z�=mz+d with ��z�=��z� as shown in

Fig. 1(b). The model adopted for the material param-

0146-9592/09/071015-3/$15.00 ©

eters allow m and d to be complex and enables an ex-act analytical solution to Maxwell’s equations to beobtained. The results we present are therefore notlimited by any computational restrictions or arti-facts.

In [11] a transitional layer between half-planes ofRHM and LHM was studied. The solution led to lo-calization of the field within the layer, and it wasfound that the inclusion of a discontinuity, �, in therefractive index at a point within the layer enabledthe GRIN model to more flexibly allow for surface de-fects and rapid changes in the refractive index. Thismodel has also recently been investigated by anothergroup [13]. However, the semi-infinite half-planemodel, while illustrating the novel effects introducedby the layer, permits evanescent modes to grow with-out bound in the LHM, and it is therefore necessaryto consider a finite thickness of LHM.

We assume that an s-polarized wave is incidentupon the slab from the left as shown in Fig. 1(a), andthe solution to Maxwell’s equations in each of the re-gions has the form E=E�z�exp�i�kxx−�t��y,

E�z� =�e�ikz1

z� + re�−ikz1z� z � − h − ah

E1�z� − h − ah � z � − h

E2�z� − h � z � − h + ah

se�ikz2z� + qe�−ikz2

z� − h + ah � z � h − ah,

E3�z� h − ah � z � h

E4�z� h � z � h + ah

te�ikz1z� h + ah � z

��1�

where q, r, s, and t are dependent on the boundaryconditions, kz1

and kz2are the z components of the

wavenumbers outside and inside the lens, respec-tively, and E1/2/3/4�z� are the electric fields in the dif-fuse layers that are expressible in terms of hypergeo-metric functions as detailed in [11].

The slab is illuminated by a planar source locatedat z=−2h with a phase function of the form

exp�i�kxx+kzz��. Such waves form the eigenmodes of

2009 Optical Society of America

1016 OPTICS LETTERS / Vol. 34, No. 7 / April 1, 2009

more complicated sources, for example, multipoles.The problem can be defined entirely by the followingnondimensional variables: the wavenumber from thesource, kh= �kx

2+kz2�1/2h; the wavenumber directed

parallel to the lens, kxh; the layer half-widths, a; thelosses in the LHM, �; and the discontinuity size, �.The two layer sizes and discontinuities need not havethe same value, but making them equal simplifiesthe problem without losing the novel effects intro-duced by the layers.

When kx�k the incoming wave is evanescent inthe z direction, and a typical field distribution fromthe source point to the primary focus at z=2h isshown in Fig. 2. Field localizations can be seen ineach layer. These have larger peak values than wouldbe anticipated by continuation of the evanescentmodes from the source (as shown by the dashedlines). The enhancement is due to the reduction inRe�n� within the layer magnifying the amplification(or decay) of evanescent waves, see [11].

The gradient of the evanescent field is also en-hanced in the layers. Consequently the most local-

Fig. 1. Illustration of (a) physical setup and (b) refractiveindex profile for a lens with diffuse boundaries.

Fig. 2. (Color online) Typical field profile across the lensnormalized against �E� for the incoming wave at z=−2h.Here �=10−5, �=0.01, a=0.2, kh=10, and kxh=10.05. Thedotted vertical lines denote the GRIN layers at the surfaces

of the lens.

ized modes, corresponding to larger values of kxh,will be dissipated if the LHM is lossy. It can thereforebe anticipated that resolution will be adversely af-fected by the presence of the layers as demonstratednext.

A measure of the superresolving capability of alens for a fixed energy kh is the maximum value ofkxh that can be accommodated at the focus. Fixing khand allowing kxh to vary from 0 to � allows the studyof the full range of propagating and evanescentwaves. Figure 3(a) shows the relationship betweenkxh and kzh. If the lens is lossless, the presence of thelayers has no effect on its perfect resolving ability.However, if the LHM has any finite (positive) value ofthe loss, the layers serve to deteriorate the lens per-formance by a margin in excess of that caused by dis-sipation alone. This is illustrated in Fig. 3(b), whichshows the transmission plotted as a function of kxhfor different values of a, where the transmission isdefined as the amplitude of the field at the focus nor-malized by that at the source.

The curves in Fig. 3(b) annotated with circles,squares, and triangles correspond to layer widths a=0, 0.01, and 0.05, respectively. For kx�k, the wavesare propagating, and the transmission falls to zero asthe modes approach grazing incidence, beyond whichthey are purely evanescent. Evanescent modes aretransmitted by the lens, but the efficiency decreaseswith increasing kxh. The effect of a thicker layer is to

Fig. 3. (Color online) (a) Real (triangle) and imaginary(square) parts of kzh and (b) transmission against kxh fora=0 (circles), 0.01 (squares), and 0.05 (triangles), �=10−2,hk=1, and �=0.01. Also shown in (b) by the dashed curve is

a=0 and �=0.08.

April 1, 2009 / Vol. 34, No. 7 / OPTICS LETTERS 1017

reduce the maximum transmission and the value ofkxh for which the transmission coefficient becomessmall. The dotted curve shows the transmission coef-ficient for a lossy ��=0.08� LHM slab without layers�a=0�. Comparing this curve with that marked by tri-angles, it can be seen that an LHM with �=0.01 andlarge layer widths is outperformed by an LHM with alarger loss ��=0.08� but with smooth surfaces.

Figure 4 shows the dependence of transmission onthe diffuse layer width for different values of the loss.For losses of the order 10−4 or more, the presence of asmall transitional layer causes a large drop in trans-mission but further increases in layer width have adiminishing effect. This is reminiscent of the loga-rithmic dependence exhibited by perturbations to thematerial parameters of LHM as shown in [9,14,15].

To put the results of Fig. 4 into perspective, currentLHMs have figures of merit �FOMs= �Re�n� / Im�n����20, making even the lowest (squares) curve (forwhich FOM=100, �=10−2) very optimistic. Hence un-til such a time that FOMs of 105–106 are attained,the quality of a lens’ boundaries will be vitally impor-tant to the recovery of evanescent modes.

This Letter has investigated the effect of imperfectboundaries upon an LHM lens using a GRIN modelfor the material parameters within transitional lay-ers located at its surfaces. The resolving capacity ofthe lens has been quantified in terms of the layerthickness. If the LHM is lossless, then the layershave no effect on the lens’ ability to resolve perfectly.

Fig. 4. (Color online) Transmission against a for FOM�=1/��=107 (triangles), 105 (circles), 104 (diamonds), and

3

10 (squares), kxh=10.3, kzh=1.95i, and �=0.01.

This is not the case if the LHM is lossy; the conse-quence of the localization caused by the layers is topreferentially dissipate high kxh modes. The detri-mental effect of this upon resolution is similar to anonlinearly enhanced value of loss in the bulk LHM.

The results show that surface imperfections willhave a substantial impact on the recovery of bothpropagating and, especially, evanescent modes. Thiswill become more important as the losses in manufac-tured LHM decrease through improved materials-science techniques. This is because the physics gov-erning the operation of the perfect lens is singular innature, and even a small layer serves to move thesystem away from resonance, leading to substantialfalls in system performance. Indeed, the deteriora-tion in superresolving power due to diffuse surface ef-fects will prove to be an exacting deficiency to over-come, even if values of dissipation can be achievedthat are orders of magnitude smaller than those cur-rently attainable.

P. Ingrey is funded by the Engineering and Physi-cal Sciences Research Council (EPSRC).

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