The asymmetric lossy near-perfect lensxlab.me.berkeley.edu/MURI/publications/publications_4.pdf · 2. The asymmetric slab Typically we consider the asymmetric lens to be a slab of

The asymmetric lossy near-perfect lens

S ANANTHA RAMAKRISHNA J B PENDRY

The Blackett Laboratory Imperial College London SW7 2BZ UKe-mails saramakrishnaicacuk jpendryicacuk

D SCHURIG D R SMITH and S SCHULTZ

Department of Physics University of California San Diego9500 Gilman Drive La Jolla California 92093-0319 USA e-maildrsucsdedu

(Received 26 September 2001 revision received 19 November 2001)

Abstract We extend the ideas of the perfect lens recently proposed [JBPendry Phys Rev Lett 85 3966 (2000)] to an alternative structure We showthat a slab of a medium with negative refractive index bounded by media ofdi erent positive refractive index also amplireges evanescent waves and can act as anear-perfect lens We examine the role of the surface states in the ampliregcation ofthe evanescent waves The image resolution obtained by this asymmetric lens ismore robust against the e ects of absorption in the lens In particular we studythe case of a slab of silver which has a negative dielectric constant with air onone side and other media such as glass or GaAs on the other side as an`asymmetricrsquo lossy near-perfect lens for p-polarized waves It is found thatretardation has an adverse e ect on the imaging due to the positive magneticpermeability of silver but we conclude that subwavelength image resolution ispossible in spite of it

1 Introduction

The electromagnetic radiation emitted or scattered by an object consists of aradiative component of propagating modes and a near-regeld component of non-propagating modes whose amplitudes decay exponentially with distance from thesource For a monochromatic planar source the electromagnetic regeld in free spacecan be expressed as a Fourier sum over all wave-vectors

Ehellipx y z tdagger ˆX

kx ky

Ehellipkx ky kzdagger exp permilihellipkxx Dagger kyy Dagger kzz iexcl tdaggerŠ hellip1dagger

where k2x Dagger k2

y Dagger k2z ˆ 2=c2 and c is the speed of light in free space For

k2x Dagger k2

y gt 2=c2 kz is seen to be purely imaginary and the case is similiar for kx

and ky The near regeld consists of these partial waves with imaginary wave-vectorsand decays exponentially away from the source These are the high-frequencyFourier components describing the rapidly varying spatial features on the objectand are never detected in conventional imaging using conventional lenses Thislack of information results in the limitation to conventional imaging that sub-wavelength features of a source cannot be resolved in the image In order to

Journal of Modern Optics ISSN 0950plusmn0340 printISSN 1362plusmn3044 online 2002 Taylor amp Francis Ltdhttpwwwtandfcoukjournals

DOI 10108009500340110120950

journal of modern optics 2002 vol 49 no 10 1747plusmn1762

overcome this limitation scanning near-regeld optical microscopy (SNOM) wasproposed where the near regeld of the radiating object is probed by bringing atapered regbre tip very close to the object (see [1] for a recent review) A rule of thethumb is that the near regeld produced by a (periodic) feature of spatial extent d willdecay exponentially at a rate of d=2p away from the surface Hence there is a needto get close to the surface at these resolutions just in order to detect the evanescentregeld Impressive advances have been made in this regeld but there still remainseveral problems in understanding the images phase contrast mechanisms andassociated artefacts

Now in order to reconstruct the object completely we would need to amplifythese evanescent waves and to give the appropriate phase shift for the propagatingcomponents This is precisely what the recently proposed perfect lens withnegative refractive index accomplishes [2] Veselago [3] had noted a long timeago that owing to the reversal of Snellrsquos law a slab of negative refractive indexwould act as a lens in that the rays from a source on one side would becomerefocused on the other side to form an image but the ampliregcation of evanescentwaves by a slab of negative refractive index noted by Pendry [2] was a surprisingand new result The possibility of negative refractive media has already beenexperimentally demonstrated in the microwave region of the spectrum [4 5] Boththe dielectric constant and the magnetic permeability are negative in a negativerefractive medium and the electromagnetic waves in such a medium will be lefthanded as a consequence of Maxwellrsquos equations No known natural materials havenegative magnetic permeability and the negative refractive media are meta-materials consisting of interlaced periodic arrays of split ring resonators [6] andthin wires [7] Pendry also showed that a thin slab of silver with negative dielectricconstant would act as a near-perfect lens for near-regeld imaging with p-polarizedwaves in the electrostatic limit with the resolution limited only by absorption inthe lens (silver) More recently we have examined some consequences of devi-ations in the dielectric constant and magnetic permeability from the perfect lensconditions (deg ˆ iexcl1 and middot ˆ iexcl1) on the resolution of the lens [8] and found that therestrictions on deg and middot were quite severe but achievable by current-day technologyThe main advantage of this near-regeld perfect lens over the current methods ofSNOM is that a complete image is generated at the image plane This would beimportant in many applications for example in-situ imaging of biological mol-ecules and processes by macruorescence imaging

Here we extend the ideas of Pendryrsquos original work We show that an asym-metric lens consisting of a slab of negative refractive index bounded by media ofdi erent positive refractive index can also act as a near-perfect lens We thenexamine the nature of the surface modes which are responsible for the ampliregca-tion of the evanescent waves In particular we study the case of a reglm of silver(with negative dielectric constant) deposited on other media such as glass or GaAswith positive dielectric constant as a near-regeld imaging lens at optical frequenciesWe show that the asymmetry in the system can actually enhance the resolutiondepending on the choice of asymmetry and the operating frequency One of themain advantages of this asymmetric lens is that a very thin metal reglm deposited ona solid substrate (glass or GaAs) will be mechanically much more stable than afree-standing metal reglm We also study the e ects of retardation on the system andconclude that subwavelength resolution is possible in spite of the adverse e ects ofretardation and absorption

1748 S A Ramakrishna et al

2 The asymmetric slabTypically we consider the asymmetric lens to be a slab of negative refractive

medium of thickness d dielectric constant deg2 and magnetic permeability middot2

between media of di ering refractive indices (reggure 1) We shall consider medium1 on one side of the source to be air (deg1 ˆ 1 and middot1 ˆ 1) and some other dielectric ormagnetic material (deg3 and middot3) on the other side We will consider the object planeto be in medium 1 at a distance d=2 and the image plane where we detect theimage inside medium 3 at a distance d=2 from the edge of the slab Let p-polarizedlight be incident on the slab from medium 1 with the magnetic regeld given by

H1p ˆ exphellipikhellip1daggerz z Dagger ikxx iexcl itdagger hellip2dagger

where k2x Dagger khellipjdagger

z

2 ˆ degjmiddot2=c2 (j ˆ 1 2 3 for the di erent media) We shall work intwo dimensions for reasons of simplicity If the index of refraction is negative(deg2 lt 0 and middot2 lt 0) then Maxwellrsquos equations and causality demand thatkhellip2dagger

z ˆ iexclhellipdeg2middot22=c2 iexcl k2xdagger1=2 if kx lt hellipdeg2middot2dagger1=2=c and khellip2dagger

z ˆ ihellipk2x iexcl deg2middot22=c2dagger1=2 if

kx gt hellipdeg2middot2dagger1=2=c [2] The transmission coe cient for the p-polarized light acrossthe slab is

Tphellipkxdagger ˆ4hellipkhellip1dagger

z =deg1daggerhellipkhellip2daggerz =deg2dagger exphellipikhellip2dagger

z ddaggerhellipkhellip1dagger

z =deg1 Dagger khellip2daggerz =deg2daggerhellipkhellip2dagger

z =deg2 Dagger khellip3daggerz =deg3dagger iexcl hellipkhellip1dagger

z =deg1 iexcl khellip2daggerz =deg2daggerhellipkhellip3dagger

z =deg3 iexcl khellip2daggerz =deg2dagger exp hellip2ik

hellip2daggerz ddagger

hellip3dagger

We immediately see that if either deg2 ˆ iexcldeg1 and middot2 ˆ iexclmiddot1 or deg2 ˆ iexcldeg3 and

middot2 ˆ iexclmiddot3 then Tp ˆ permil2deg3khellip1daggerz =hellipdeg1khellip3dagger

z Dagger deg3khellip1daggerz daggerŠ exphellipiexclikhellip2dagger

z ddagger and the ampliregcationof the evanescent waves as well as the phase reversal for the propagating com-

Asymmetric lossy near-perfect lens 1749

It is more convenient to use the magnetic regeld for the p-polarized light The electricregeld can be obtained by using Maxwellrsquos equation ~kkhellipjdagger pound ~BB ˆ iexclj

~EE

1 2 3

r32

r 12

t 32t 21

d2

Medium-3SilverAir

d d2

Figure 1 The asymmetric lens system consisting of a slab of negative refractive indexbounded by air on one side and any positive medium on the other The object inmedium 1 at a distance d=2 from the silver surface is focused inside medium 3 Thepartial remacrection and transmission coe cients across the boundaries are shown

ponents result Thus the system does not have to be symmetric (deg1 ˆ deg3 andmiddot1 ˆ middot3) as in the original work of Pendry for accomplishing ampliregcation ofevanescent waves In this asymmetric case however the regeld strength at the imageplane di ers from the object plane by a constant factor and thus the imageintensity is changed Further it is easily verireged that wave-vectors with di erentkx will refocus at slightly di erent positions unless deg3 ˆ deg1 and middot3 ˆ middot1 (ie thesymmetric case) or in the limit kx 1 Thus there is no unique (perfect) imageplane and we should expect the image to su er from aberrations Hence we termthe asymmetric slab a near-perfect lens A similiar result holds for the s-polarizedwave incident on the slab Again we point out that with this asymmetric lens theobject is considered to be in medium 1 (usually air) and the image is formed insidemedium 3 (mostly considered to be a high index dielectric in subsequent sections ofthis paper)

In the electrostatic (magnetostatic) approximation (kx 1) we havekhellipjdagger

z ikx and the dependence of the transmission coe cient for the p(s)-polarizedregeld on middot(deg) is eliminated The transmission coe cient for the p-polarized waveacross the slab is then given by

Tphellipkxdagger ˆ 4deg2deg3 exp hellipiexclkxddaggerhellipdeg1 Dagger deg2daggerhellipdeg2 Dagger deg3dagger iexcl hellipdeg2 iexcl deg1daggerhellipdeg2 iexcl deg3dagger exphellipiexcl2kxddagger hellip4dagger

and the ampliregcation of the evanescent waves depends only on the condition onthe dielectric constant (deg2 ˆ iexcldeg1 or deg2 ˆ iexcldeg3) Such a system can be easilyrealized as several metals act as a good plasma in some range of optical frequenciesin that they can have a large negative real part of the dielectric constant witha comparatively small imaginary part Typically we take our system to be a thinreglm of silver deposited on another medium such as glass GaAs or silicon We notethat silver is highly dispersive and by choosing the operating wavelength of lightappropriately the dielectric constant deg2 of silver can be chosen to be either iexcldeg1 oriexcldeg3 Note that middot ˆ Dagger1 everywhere for such a system In the electrostatic limit wecan take the object plane and the image plane to be symmetric about the slab ata distance d=2 from the edge of the slab To arrive at a simple althoughapproximate description of the asymmetric lens we shall take the electrostaticlimit in the remainder of this section We shall consider the e ects of retardation insection 4

The link between the ampliregcation of the evanescent waves to the presence of asurface plasmon mode has already been pointed out [2] To obtain an insight intothe process let us examine the spatial regeld variation in our system First let usconsider the transmission T and the remacrection R from the slab as a sum of partialwaves arising from multiple scattering at the interfaces

T ˆ t21t32 exphellipiexclkxddagger Dagger t21r32r12t32 exphellipiexcl3kxddagger Dagger t21r32r12r32r12t32 exphellipiexcl5kxddaggerDagger cent cent cent

ˆ t21t32 exphellipiexclkxddagger1 iexcl r32r12 exphellipiexcl2kxddagger hellip5dagger

R ˆ r21 Dagger t21r32t12 exphellipiexcl2kxddagger Dagger t21r32r12r32t12 exphellipiexcl4kxddagger Dagger cent cent cent

ˆ r21 Dagger t21r32t12 exphellipiexcl2kxddagger1 iexcl r32r12 exphellipiexcl2kxddagger hellip6dagger


where tjk ˆ 2degj=hellipdegk Dagger degjdagger and rjk ˆ hellipdegj iexcl degkdagger=hellipdegk Dagger degjdagger are the partial transmissionand remacrection Fresnel coe cients for p-polarized light (obtained by matching thetangential components of E and H) across the interface between medium j andmedium k (see reggure 1) When the perfect-lens condition at any of the interfaces issatisreged the partial remacrection coe cient rjk as well as the partial transmissioncoe cient tjk for evanescence across the interface diverge It should be notedhowever that the S matrix that relates the incoming and the outgoing waveamplitudes in the scattering process is analytic in the complex (momentum orenergy) plane Hence the sum of the inregnite series is still valid owing to theanalyticity of the S matrix and the remacrection and the transmission from the slabare well-deregned quantities The above procedure is entirely equivalent to the usualtime-honoured practice of decomposing the regelds into a complete set of basis statesand matching the regeld amplitudes at the boundaries as follows Let the regelds inregions plusmn 1 2 and 3 be given by

H1 ˆ exphellipiexclkxzdagger Dagger R exphellipkxzdagger hellip7dagger

H2 ˆ A exphellipiexclkxzdagger Dagger B exphellipkxzdagger hellip8dagger

H3 ˆ T exphellipiexclkxzdagger hellip9dagger

where R and T represent the remacrection and transmission coe cients respectively ofthe slab Matching the tangential components of H and E at the boundaries weobtain after some trivial algebra

Ahellipdeg2 ˆ iexcldeg1dagger ˆ deg3 iexcl deg1

deg3 Dagger deg1

exphellip2kxddagger Ahellipdeg2 ˆ iexcldeg3dagger ˆ 0 hellip10dagger

Bhellipdeg2 ˆ iexcldeg1dagger ˆ 1 Bhellipdeg2 ˆ iexcldeg3dagger ˆ2deg3

deg3 Dagger deg1 hellip11dagger

Thellipdeg2 ˆ iexcldeg1dagger ˆ 2deg3

deg3 Dagger deg1

exphellipkxddagger Thellipdeg2 ˆ iexcldeg3dagger ˆ 2deg3

deg3 Dagger deg1

exphellipkxddagger hellip12dagger

Rhellipdeg2 ˆ iexcldeg1dagger ˆ deg3 iexcl deg1

deg3 Dagger deg1

exphellipDagger2kxddagger Rhellipdeg2 ˆ iexcldeg3dagger ˆ deg3 iexcl deg1


Note that there are two separate cases deg2 ˆ iexcldeg1 on the left-hand side and deg2 ˆ iexcldeg3

on the right-hand side in the above equationsFirstly we note that while the transmissions are the same in both the cases

the spatial variations in the regeld are completely di erent as shown in reggure2 Secondly there is a non-zero remacrection in both cases and for deg2 ˆ iexcldeg1

the remacrection is also amplireged The regrst case of deg2 ˆ iexcldeg1 is exactly the conditionfor the excitation of a surface plasmon state at the airplusmnsilver interface wherethe regeld is seen to be large and decaying on either side of the interface The othercase of deg2 ˆ iexcldeg3 corresponds to the excitation of a surface plasmon state atthe silverplusmnGaAs interface In this case only the growing solution within the silverslab is present It is interesting to note that in the symmetric case when deg1 ˆ deg3again only this growing solution within the slab is selectively excitedThe remacrectivity is also zero for the symmetric slab To understand the roleof the surface modes in the ampliregcation of the evanescent regeld we notethat the condition deg2 ˆ iexcldeg13 is exactly that for a surface plasmon state toexist on an interface between the semi-inregnite negative and semi-inregnite positive


media In a regnite slab with two surfaces however the surface plasmons detunefrom the resonant frequency and the detuning is exactly of the right magnitude toensure that they are excited by incident regelds to the correct degree for a focusedimage

The presence of a large remacrectivity has serious consequences for the use of alens for near regeld imaging applications as it would disturb the object regeld We callthe regrst case when deg2 ˆ iexcldeg1 where the remacrection is also amplireged the unfavour-able conregguration and the second case when deg2 ˆ iexcldeg3 the favourable conreggurationof the asymmetric lens Obviously it would be most advantageous if the remacrectivitywere zero Here we draw an analogy with a conventional lens which in the ideal hasperfectly transmitting surfaces but in practice always produces some stray remacrec-tion from the lens surface but nevertheless still provides an acceptable degree offunctionality

3 E ects of absorptionNow we shall consider the e ect of absorption in the slab of the negative-

index medium on the imaging process We shall work in the electrostatic limit orassume that the perfect lens conditions for the magnetic permeability (middot2 ˆ iexclmiddot1 or


-10 00 10 20 3005

15

25

10

20

30

4020

30

40

Object plane Silver slab Image plane

e = -e

e = -e

e = e = -e

1 2

3 2

1 3 2

Figure 2 Schematic diagram of the regeld strength distribution for (a) the asymmetricslab with the surface plasmon excited at the airplusmnsilver interface (b) the asymmetricslab with the surface plasmon excited at the interface between silver and the higher-dielectric-constant medium and (c) the symmetric slab with air on both sides Theregeld strength at the image plane is the same as the object plane in this case

middot2 ˆ iexclmiddot3) is also satisreged Pendry [2] had shown that the eventual resolution islimited by absorption in the lens This absorption can be included in thecalculation by adding an imaginary part to the dielectric constant Pendry hadused the dispersion deg2hellipdagger ˆ 57 iexcl 902hellipmiddothdaggeriexcl2 Dagger i04 for silver (middoth in electronvolts)and we shall continue to use it here Put deg2 ˆ iexcldegk Dagger ideg0

2 where k is either 1 or 3The expression for the transmission becomes (assuming that deg0

2 frac12 degk)

Tp ˆ iexcl4degkdeg3 exphellipiexclkxddaggersectideg0

2hellipdeg1 iexcl deg3dagger iexcl 2degkhellipdeg1 Dagger deg3dagger exphellipiexcl2kxddagger hellip14dagger

with the sect being chosen according to whether k is 1 or 3 respectively Theresolution in Pendryrsquos original case was limited because for large kx the exponen-tial term in the denominator becomes smaller than the other term Now we can seethat there is deregnitely an advantage in choosing degk to be the larger of deg1 and deg3 inorder to make the term containing the exponential dominate in the denominatorThus the asymmetry can actually help us to better the limit on the resolution set byabsorption Fortunately this also corresponds to the favourable case whendeg2 ˆ iexcldeg3 Choosing degk to be the smaller of the two would in contrast causedegradation of the resolution Hence the subwavelength resolution proposedearlier by us as the ratio of the optical wavelength to the linear size of smallestresolved feature [8] works out to be

res ˆ para0

paraminˆ iexcl

ln jdeg02=2deg3jpara0

4pd hellip15dagger

(assuming that deg3 frac34 deg02 and deg3 frac34 deg1) in the favourable conregguration

Below we shall consider the above e ects of absorption on the image of twouniformly illuminated slits (intensity 1) obtained in transmission by the asym-metric slab of silver of thickness d ˆ40 nm On the other side we consider severalmedia of higher dielectric constant We numerically obtain the image at the imageplane (considered to be at d=2 in the electrostatic limit) and plot these in reggures 3(a) and (b) In reggure 3 (a) we consider the case where the resolution is enhancedowing to the asymmetry in the lens We take deg2 ˆ iexcldeg3 Dagger ideg0

2 by tuning to theappropriate frequency (lower according to the dispersion form) where deg3 gt 1 forall the media considered water (deg3 ˆ 177) glass (deg3 ˆ 225) zirconia (deg3 ˆ 46)and silicon (deg3 ˆ 149 at middothsp ˆ 2 eV corresponding to when the surface plasmon isexcited at the silverplusmnsilicon interface) The levels of absorption in silver are takento be approximately the same at all these frequencies We consider the object toconsist of two slits 20 nm wide and placed apart by a centre-to-centre distance of80 nm We regnd that the best contrast is obtained for the case of silicon rather thanthe symmetric lens In fact the resolution initially degrades with increasing deg3 butimproves drastically for large values of deg3 In this case the two slits are notresolved for the case of glass and zirconia We obtain a limit of the spatialresolution of about 70 nm (centre to centre) for the case of silicon In reggure 3 (b)we consider the images obtained for the case deg2 ˆ iexcldeg1 Dagger ideg0

2 where deg1 ˆ 10 for airon one side This is the unfavourable conregguration corresponding to the surfaceplasmon excitation (middothsp ˆ 348 eV) at the airplusmnsilver interface We consider a largerobject of two slits of width 30 nm the centres placed apart by 120 nm and regnd thatthe image resolution becomes degraded with increasing deg3 For the case of silicon(deg3 ˆ 164 at this frequency) the image is just resolved according to the Rayleigh


criterion for resolution However we shall not be particularly interested in thiscase owing to the amplireged remacrected wave which will severely a ect the imaging

4 E ects of retardationIn the electrostatic(magnetostatic) limit of large kx ordm kz ordm qz there is no e ect

of changing middot(deg) for the p(s) polarization The deviation from the electrostatic limit


It should be noted that the Rayleigh criterion in the strict sense is applicable for theradiative modes It states that two separate sources are just resolved when the principalmaximum of one image coincides with the regrst minimum in the di raction pattern of theother This results in the ratio of the intensity at the midpoint between the sources to theintensity at the maxima of about 0811 We shall use this ratio as the equivalent measure forderegning the resolution in our case

-1500 -1000 -500 00 500 1000 1500X (nm) --gt

000

005

010

015

020

025

030 Water (18)Air (10)Glass (23)Zirconia (462)Silicon (164)

-1500 -1000 -500 00 500 1000 1500000

010

020

030

040Air (10)Water (177)Glass (225)Zirconia (46)Silicon (149)

(a) enhanced resolution

(b) degraded resolution

Figure 3 The images (intensity at the image plane in arbitrary units) obtained in theelectrostatic approximation for the favourable and unfavourable conreggurationsshowing the e ects of absorption in the slab on the image resolution The positionsof the slits which are imaged are shown by dotted lines The dielectric constants ofthe media are shown within parentheses in the keys (a) Enhancement of the imageresolution caused by the asymmetric lens when deg2 ˆ iexcldeg3 Dagger i04 that is thefavourable conregguration The slits (20 nm wide and centre-to-centre spacing of80 nm) are just resolved for the symmetric case (air) but are well resolved for thelarge deg3 ˆ 149 of silicon (b) Degradation of the image resolution with increasingdielectric constant of medium 3 when deg2 ˆ iexcl1 Dagger i04 that is the unfavourableconregguration The slits 30 nm wide (larger than for (a)) with a centre-to-centrespacing of 120 nm (greater than for (a)) are barely resolved for the case of silicon

caused by the non-zero frequency of the electromagnetic wave would howevernot allow this decoupling We shall now proceed to investigate the e ects ofretardation caused by a regnite frequency of the light It has already been noted [8]that a mismatch in the deg and middot from the perfect-lens conditions would always limitthe image resolution and also leads to large transmission resonances associatedwith the excitation of coupled surface modes that could introduce artefacts into theimage

Let us regrst examine the nature of the surface plasmons (for p-polarizedincident light) at a single interface between a positive and a negative mediumThe condition for the existence of a surface plasmon at the interface is [9]

khellip1daggerz

deg1

Dagger khellip2daggerz

deg2

ˆ 0 hellip16dagger

which gives the dispersion relation

kx ˆ

c

deg2hellipdeg2 iexcl middot2daggerdeg22 iexcl 1

sup3 acute1=2

hellip17dagger

where it is assumed that deg1 ˆ 1 and middot1 ˆ 1 for the positive medium Note thatequation (16) can be satisreged only for imaginary khellip2dagger

z when deg2 is negative Thedispersion is plotted in reggure 4 where the causal plasma dispersion formsdeg2 ˆ 1 iexcl 2

p=2 and middot2 ordm 1 iexcl 2mp=2 are assumed for the material of the negative

medium [6 7] This is because a dispersionless middothellipdagger lt 0 at all would not bephysical [10] We can see that the plasmon dispersion take di erent forms for

mp gt p (the upper curve) and mp lt p (the lower curve) At large kx theplasmon frequency tends to the electrostatic limit of p=21=2 from either above for

mp gt p or below for mp lt p At small kx the surface plasmon regrst appears atthe light line ( ˆ ckx) We note that similiar results have been obtained by Ruppin[11] very recently for the dispersion of the surface plasmon modes in negativerefractive media for the case p gt mp

Now let us next consider the symmetric lossless slab (deg1 ˆ deg3 middot1 ˆ middot3 ˆ 1 and

deg02 ˆ 0) and examine the nature of these surface modes As is well known the two

degenerate surface plasmons at the two interfaces became coupled for a thin slaband give rise to coupled slab modes a symmetric and an antisymmetric mode Thecondition for these resonances (for p-polarized light) are [9]

tanhkhellip2dagger

z d

2

sup3 acuteˆ iexcl 2khellip1dagger

z

khellip2daggerz

hellip18dagger

cothkhellip2dagger

z d

2


z

khellip2daggerz

hellip19dagger

where khellipjdaggerz ˆ hellipk2

x iexcl degjmiddotj2=c2dagger1=2 Using the above conditions the dispersion rela-

tion for the coupled surface modes can be obtained and we show them in reggure 4where again we have assumed the dispersion forms deghellipdagger ˆ 1 iexcl 2

p=2 and

middot2 ˆ 1 iexcl 2mp=2 We regnd that the regnite frequency changes the dispersion from

the electrostatic limit (where it is given by ˆ ppermil1 sect exphellipiexclkxddaggerŠ1=221=2) Thedispersion relations are qualitatively di erent for mp lt p (or middot2hellipsdagger gt iexcl1) and

mp gt p (or middot2hellipsdagger lt iexcl1) In the former case there will be at least twotransmission resonances at a given for a deviation of deg2 from the perfect lens


condition In fact for small enough negative deviation deg2 ˆ iexcl1 iexcl macr (macr gt 0 butsmall) the regrst condition can be satisreged for three kx and up to four transmissionresonances are possible In the latter case when middot2 lt iexcl1 there will be at least onetransmitted resonance that can be excited at any For small enough positivedeviation deg2 ˆ iexcl1 Dagger macr (macr gt 0 but small) the second condition can be satisreged fortwo kx and up to three transmission resonances are possible This becomes clearfrom reggures 5 (b) and (d) and the transmission resonances are shown in reggures 6(a) and (c) Mathematically the reason for this behaviour can be explained asfollows For kx 1 s sect macr (where macr gt 0 and 0) the sect sign being chosendepending on whether middot2 lt iexcl1 or middot2 gt iexcl1 This causes either the lower or theupper branch respectively to intersect the ˆ s line at some point Physicallythis is because the behaviour of both the symmetric and the antisymmetric modesat large kx have to tend to the uncoupled plasmon dispersion for a single surfaceFor positive deviation in deg2 and large enough k0d it is possible that no surfacemode can be excited in either case but the required ampliregcation of the evanescentwaves also does not result


00 20 40k k

00

05

10w

w

w = 00 w = 02w w = 2w

ll

(1)

mp

mp

mp p

p

x p

p

Figure 4 The dispersion for the surface plasmon at an interface between a negative anda positive medium when the e ects of dispersion are included The frequency axis isscaled with respect to the bulk plasmon frequency p and the kx axis is scaled withrespect to kp ˆ wp=c The light line ˆ ckx and the ˆ s ˆ p=21=2 line are givenby the dotted lines The dispersion inside the inregnite negative medium ˆ ckx=hellipdeg2middot2dagger1=2

with mp ˆ 2p is shown by curve (1) (ETH plusmn ETH)

In reggure 6 we show the transfer function from the object plane to the imageplane across a slab of negative dielectric constant for the deviations of the various

parameters from the perfect lens condition In reggure 6 (a) we show the transmis-

sion for middot2 lt 1 Most values of deg2 give a single resonance However for the case of

small positive deviation (deg2 ˆ iexcl0999 98) three resonances arise the latter two

occur very close together and are not resolved in the graph We note in this case

that even a small amount of absorption removes all traces of the resonances at large

kx In the case of the p-polarized light and for middot2 lt 1 the e ect of the deviation in

middot is much lesser than that for deg (real or imaginary) Figure 6 (b) shows the

transmission for middot2 gt iexcl1 There are now at least two peaks the one which was

absent in the other case being very close to kx ˆ k0 In reggure 6 (c) we show the

transmission for middot2 ˆ 1 The additional transmission resonances at large kx for

small negative deviation in deg2 (ˆ iexcl1001) are clearly visible (expanded and shown

in reggure 6 (d)) Again we note that non-zero absorption in the slab prevents the


00 20 40 60 80 100k k

00

02

04

06

08

10

ww

k d=03light modes

00 20 40 60 80 100k k

00

02

04

06

08

10w

w

k d=03k d=05

150 250 350 450 550 650k k

07070

07071

07072

07073

07074

100 200 300 400 500k k

07068

07069

07070

07071

07072(a) (b)

(c) (d)

x p x p

p x x p

p p

p

p

p

ll

Figure 5 The dispersion for the coupled surface plasmons when the e ects ofretardation are included The frequency axis is scaled with respect to the bulkplasmon frequency p and the kx axis is scaled with respect to kp ˆ wp=c The lightline ˆ ckx and the ˆ p=21=2 line are given by the dotted lines (a) middot2 ˆ Dagger10(b) Behaviour in (a) at larger kx (c) middot ˆ 1 iexcl 2

mp=2 with mp ˆ 2p The dispersionin the inregnite negative medium ˆ ckx=hellipdeg2middot2dagger1=2

is also shown (plusmn cent plusmn) Part of thelower branch (^) shows the non-radiative light modes within the slab (d) Behaviourat larger kx in (c)

divergences at the resonances but the resonance at kx close to k0 is not as strongly

damped by absorption

The transmission resonances will cause the Fourier components at that spatial

frequency to be disproportionately represented in the image and will introduce`noisersquo in the image The slab will also act as a `low-passrsquo reglter for the spatial

frequencies because of the deviation (in the real or the imaginary parts of deg2 and middot2)

from the perfect-lens condition and limits the image resolution [8] However any

regnite amount of absorption in the slab will not allow the transmission to diverge

When the e ects of absorption is included in the calculation the transmittedintensity at the resonances will be large but regnite and we expect that the e ects of

the resonances will be softened However absorption also increases the deviation

from the perfect-lens conditions and will limit the range of spatial frequencies with

e ective ampliregcation Hence absorption by itself would not cause a fundamental

improvement in the image resolutionThe problem in using the slab of silver as a perfect lens even for p-polarized

light is that the imaging is severely a ected by the e ects of retardation arising

from large deviation in the magnetic permeability (middot ˆ 1 everywhere) from


00 100 200 300 40010

-4

10-2

100

102

e=-1001e=-0999e=-1001+i0001

00 200 400 60010-6

10-4

10-2

100

102

104

e=-100002e=-1+i00001 e= -099998

310 315 32010-2

100

102

00 100 200 300 40010-4

10-2

100

102

104

m=-05m = 0m = +1

k d =0350

m =-11k d =035

e=-100

k d=040m=10

Figure 6 The absolute value of the transmitted regeld at the image plane jT exphellipikhellip1daggerz ddaggerj

for the symmetric slab as a function of kx showing the resonances caused bydeviations in deg and middot from the perfect lens conditions The deg and middot indicated in thegraphs are for the material of the slab It is assumed that deg ˆ 1 and middot ˆ 1 outside theslab The x axis is scaled with respect to the free-space propagation vector k0

the perfect-lens condition (middot2 ˆ iexclmiddot3) As it happens this large deviation of

middot severely restricts the range of Fourier components which are e ectivelyamplireged in the silver reglm In fact the e ect of absorption other than to

soften the e ects of the transmission resonances turns out to be small in

comparison For the case of a lossless (with a very small absorption) symmetric

slab with negative dielecric constant (deg2 ˆ iexcl1 and middot ˆ 1) the plasmon frequency

is middothsp ˆ 348 eV and k0 ˆ 177 pound 10iexcl2 nmiexcl1 As can be seen in reggure 7 (a)

the e ects of retardation become severe and the image of a pair of slitsis completely swamped by the `noisersquo and a large peak develops in between

the images of the slit The positions of the peaks are also di erent However

the presence of absorption softens all the singular behaviour at the resonances and

we end up with a good image in spite of the e ects of retardation which is now

mainly to reduce the intensity of the image This is shown in reggure 7 (b) for the


-2000 -1000 00 1000 2000X (nm) --gt

0000

0005

0010

0015

0020Asymmetric lossy lens

e=-15+04i e=-18+04i

-2000 -1000 00 1000 20000000

0010

0020

0030Symmetric lossy lens

-2000 -1000 00 1000 200000

10

20

Figure 7 E ects of retardation on the images middot ˆ Dagger10 everywhere The thickness ofthe slab d ˆ 40 nm The object is a pair of slits of width 20 nm placed apart by100 nm shown by broken lines in (a) and (b) and by the dotted lines in (c) (a) Thesymmetric lossless lens (b) The symmetric lossy silver lens with middoth ˆ 348 eV (c)The asymmetric lossy silver slab with silicon as medium 3 (deg3hellipmiddoth ˆ 2 eVdagger ordm 15)The broken curve is for the case when a deviation in deg2 ˆ iexcl180 Dagger i04 compensatesfor the deviation in middot from the perfect-lens conditions

case of the symmetric lossy slab of silver in air The central lobe is very small nowand the positions of the peaks are slightly shifted Although there is signiregcantdistortion of the image the main features of the image are seen to be unalteredThus absorption has an interesting role of limiting the resolution and yet beingvital for the image formation

Now we shall consider the e ects of retardation on the asymmetric lossynegative dielectric lens with a positive magnetic permeability (middot2 ˆ Dagger1) In thiscase we have only one slab mode that is active and the dispersion for it looks likethe lower branch in reggure 5 but with a di erent value of s ˆ p=hellip1 Dagger deg3dagger1=2 Thelarge deg3 however amplireges the e ects of the large deviation of the magneticpermeability (middot ˆ 1 everywhere) from the perfect-lens conditions This drasticallyreduces the range of Fourier frequencies for which e ective ampliregcation ispossible This e ect is however o set by the beneregcial and larger e ect causedby a reduced optical frequency (middoth ˆ 2 eV) for the surface plasmon excitation atthe silverplusmnsilicon interface Thus the e ects of retardation are much reduced andwe are closer to the electrostatic limit in this case The image obtained for theasymmetric lossy lens with silicon on the image side (k0 ˆ 102 pound 10iexcl2 nmiexcl1

corresponding to middoth ˆ 2 eV) is shown in reggure 7 (c) The e ects of retardationon the image in this case are seen to be much smaller and the image is not `noisyrsquoalthough the intensity is much reduced owing to the lower transmission comparedwith the electrostatic case Note that we are able to resolve subwavelength featuresin this case at about para=10 length scale This should be compared with the case ofthe symmetric lens where a resolution of only about para=4 is obtained but at asmaller wavelength

Finally we explore the possibility that a deviation of deg2 from the perfect-lenscondition may be used to compensate for the deviation of middot2 in order to obtain abetter image We show the transmission across the asymmetric slab in reggure 8 fordi erent deg2 while keeping deg1 ˆ 1 for air and deg3 ˆ Dagger15 for silicon regxed This canagain be achieved by tuning the frequency of light and using the materialdispersion of silver We note that the slab mode can be excited by making deg2

more negative (ˆ iexcldeg3 iexcl macr where macr gt 0) but the transmission remains regnite owingto the regnite absorption even at the resonant kx The transmission is increasedwithin a regnite window of kx for such a deviation even though the transmissiondecays much faster at higher values of kx frac34 k0 This is clearly seen in reggure 8 forthe cases deg2hellipmiddoth ˆ 189 eVdagger ˆ iexcl17 Dagger i04 and deg2hellipmiddoth ˆ 185 eVdagger ˆ iexcl18 Dagger i04 Adeviation of deg2 on the other side (making it more positive) just reduces thetransmission even further Note that the large transmission for kx close to k0 isnot due to a resonance of the system in this case Thus we can obtain some extratransmission at lower (but subwavelength) spatial frequencies at the cost ofreduced transmission at higher spatial frequencies The enhancement in theimage features obtained by such a compensation for the deviation in middot by changing

deg2 from the perfect-lens condition is shown in reggure 7 (c) by the broken curve Theenhanced resolution is clearly visible

5 ConclusionsIn conclusion we have shown that a slab of negative refractive index embedded

between two media of di erent (positive) refractive indices also behaves as a near-perfect lens whose performance is limited by only the e ects of absorption and


retardation It is only necessary that one of the interfaces satisreges the conditions forthe existance of a surface mode at the operating frequency We have shown that theasymmetry can be used to enhance the resolution of the lens against the e ects ofabsorption in the slab In particular we examined the case of a negative dielectriclens consisting of a slab of silver with air on one side and any other dielectric mediumon the other We found that retardation e ects due to a regnite frequency of light andthe mismatch in the magnetic permeability (middot ˆ 1 everywhere) limit the imageresolution severely and also result in the excitation of slab resonances that degradethe performance of the lens Absorption alleviates the e ects of the sharp resonancesto some extent and indeed is vital to the imaging itself The reduced frequency oflight corresponding to the surface plasmon excitation at the interface between silverand high-dielectric-constant medium can reduce the adverse e ects of the retardationdrastically We have shown that subwavelength imaging at optical frequencies(middoth ordm 20 eV) is possible by our asymmetric lens consisting of a slab of silverbounded by air on one side and silicon or GaAs on the other and we obtain aspatial resolution of about 60plusmn70 nm with this system

The possibility of having a medium to support a thin reglm of silver permits theconstruction of a nanoscopic mechanically rugged lens for optical near-regeldimaging It also admits the possibility of an integrated detection system by placingfor example an array of quantum dots at the image plane inside the GaAs


10 50 10 150k k

10-3

10-2

10-1

100

101

|T e

xp[-

(k

+k

) d

2]|

e =-15+04ie =-17+04ie =-18+04ie =-14+04i

2

(1)

2

2

2

(3)

z

k d = 035

x 0

z

0

Figure 8 The absolute value of the transmitted regeld at the image plane across theasymmetric slab for di erent deg2 deg1 ˆ 10 and deg3 ˆ 150 are kept regxed

(medium 3) It would also be easier to achieve better surface uniformities whichwill eventually limit the performance of the lens One more advantage is that theoperating frequency is reduced from middoth ˆ 348 eV in the ultraviolet for thesymmetric silver slab in air to middoth ordm 2 eV in the visible for the asymmetric lensThe advantages of this reduction of operating frequency from the ultraviolet to thevisible for optical instrumentation are obvious The one comparative disadvantageof the asymmetric lens is a non-zero remacrection coe cient which can introduceartefacts in the image This is because the remacrected regeld can now be multiplyscattered between the source and the lens to disturb the object regeld This ishowever a problem with the current methods of SNOM as well It might becomenecessary to account for the remacrection in order to obtain good images

Acknowledgments

The authors ackowledge support from the DoDONR MURI grant N00014-01-1-0803

References[1] Greffet J-J and Carminati R 1997 Prog Surf Sci 56 133[2] Pendry J B 2000 Phys Rev Lett 85 3966[3] Veselago V G 1968 Soviet Phys Uspekhi 10 509[4] Smith D R Padilla W J Vier D C Nemat-Nasser S C and Schultz S

2000 Phys Rev Lett 84 4184[5] Shelby R A Smith D R and Schultz S 2001 Science 292 77[6] Pendry J B Holden A J Robbins D J and Steward W J 1999 IEEE Trans

Microwave Theory Techniques 47 2075[7] Pendry J B Holden A J Stewart W J and Youngs I 1996 Phys Rev Lett

76 4773 Pendry J B Holden A J Robbins D J and Steward W J 1998 JPhys condens Matter 10 4785

[8] Smith D R Schurig D Rosenbluth M Schultz S Ramakrishna S A andPendry J B 2001 unpublished

[9] Raether H 1988 Surface Plasmons (Berlin Springer) Raether H 1980 Excitationof Plasmons and Interband Transitions by Electrons (Berlin Springer)

[10] Landau L D and Lifshitz E D 1960 Electrodynamics of Continuous Media(Oxford Pergamon)

[11] Ruppin R 2000 Phys Lett A 277 61 2001 J Phys condens Matter 13 1811

1762 Asymmetric lossy near-perfect lens

overcome this limitation scanning near-regeld optical microscopy (SNOM) wasproposed where the near regeld of the radiating object is probed by bringing atapered regbre tip very close to the object (see [1] for a recent review) A rule of thethumb is that the near regeld produced by a (periodic) feature of spatial extent d willdecay exponentially at a rate of d=2p away from the surface Hence there is a needto get close to the surface at these resolutions just in order to detect the evanescentregeld Impressive advances have been made in this regeld but there still remainseveral problems in understanding the images phase contrast mechanisms andassociated artefacts

Now in order to reconstruct the object completely we would need to amplifythese evanescent waves and to give the appropriate phase shift for the propagatingcomponents This is precisely what the recently proposed perfect lens withnegative refractive index accomplishes [2] Veselago [3] had noted a long timeago that owing to the reversal of Snellrsquos law a slab of negative refractive indexwould act as a lens in that the rays from a source on one side would becomerefocused on the other side to form an image but the ampliregcation of evanescentwaves by a slab of negative refractive index noted by Pendry [2] was a surprisingand new result The possibility of negative refractive media has already beenexperimentally demonstrated in the microwave region of the spectrum [4 5] Boththe dielectric constant and the magnetic permeability are negative in a negativerefractive medium and the electromagnetic waves in such a medium will be lefthanded as a consequence of Maxwellrsquos equations No known natural materials havenegative magnetic permeability and the negative refractive media are meta-materials consisting of interlaced periodic arrays of split ring resonators [6] andthin wires [7] Pendry also showed that a thin slab of silver with negative dielectricconstant would act as a near-perfect lens for near-regeld imaging with p-polarizedwaves in the electrostatic limit with the resolution limited only by absorption inthe lens (silver) More recently we have examined some consequences of devi-ations in the dielectric constant and magnetic permeability from the perfect lensconditions (deg ˆ iexcl1 and middot ˆ iexcl1) on the resolution of the lens [8] and found that therestrictions on deg and middot were quite severe but achievable by current-day technologyThe main advantage of this near-regeld perfect lens over the current methods ofSNOM is that a complete image is generated at the image plane This would beimportant in many applications for example in-situ imaging of biological mol-ecules and processes by macruorescence imaging

Here we extend the ideas of Pendryrsquos original work We show that an asym-metric lens consisting of a slab of negative refractive index bounded by media ofdi erent positive refractive index can also act as a near-perfect lens We thenexamine the nature of the surface modes which are responsible for the ampliregca-tion of the evanescent waves In particular we study the case of a reglm of silver(with negative dielectric constant) deposited on other media such as glass or GaAswith positive dielectric constant as a near-regeld imaging lens at optical frequenciesWe show that the asymmetry in the system can actually enhance the resolutiondepending on the choice of asymmetry and the operating frequency One of themain advantages of this asymmetric lens is that a very thin metal reglm deposited ona solid substrate (glass or GaAs) will be mechanically much more stable than afree-standing metal reglm We also study the e ects of retardation on the system andconclude that subwavelength resolution is possible in spite of the adverse e ects ofretardation and absorption







z













hellip3dagger







~EE

1 2 3

r32

r 12

t 32t 21

d2

Medium-3SilverAir

d d2



















deg3 Dagger deg1





deg3 Dagger deg1


deg3 Dagger deg1



deg3 Dagger deg1













-10 00 10 20 3005

15

25

10

20

30

4020

30

40


e = -e

e = -e

e = e = -e

1 2

3 2

1 3 2




2 frac12 degk)




res ˆ para0

paraminˆ iexcl


4pd hellip15dagger











-1500 -1000 -500 00 500 1000 1500X (nm) --gt

000

005

010

015

020

025


-1500 -1000 -500 00 500 1000 1500000

010

020

030







khellip1daggerz

deg1


deg2

ˆ 0 hellip16dagger


kx ˆ

c


sup3 acute1=2

hellip17dagger










tanhkhellip2dagger

z d

2


z

khellip2daggerz

hellip18dagger

cothkhellip2dagger

z d

2


z

khellip2daggerz

hellip19dagger




p=2 and







00 20 40k k

00

05

10w

w

w = 00 w = 02w w = 2w

ll

(1)

mp

mp

mp p

p

x p

p

















00 20 40 60 80 100k k

00

02

04

06

08

10

ww

k d=03light modes

00 20 40 60 80 100k k

00

02

04

06

08

10w

w

k d=03k d=05

150 250 350 450 550 650k k

07070

07071

07072

07073

07074

100 200 300 400 500k k

07068

07069

07070

07071

07072(a) (b)

(c) (d)

x p x p

p x x p

p p

p

p

p

ll



















00 100 200 300 40010

-4

10-2

100

102

e=-1001e=-0999e=-1001+i0001

00 200 400 60010-6

10-4

10-2

100

102

104

e=-100002e=-1+i00001 e= -099998

310 315 32010-2

100

102

00 100 200 300 40010-4

10-2

100

102

104

m=-05m = 0m = +1

k d =0350

m =-11k d =035

e=-100

k d=040m=10















-2000 -1000 00 1000 2000X (nm) --gt

0000

0005

0010

0015


e=-15+04i e=-18+04i

-2000 -1000 00 1000 20000000

0010

0020


-2000 -1000 00 1000 200000

10

20














10 50 10 150k k

10-3

10-2

10-1

100

101

|T e

xp[-

(k

+k

) d

2]|

e =-15+04ie =-17+04ie =-18+04ie =-14+04i

2

(1)

2

2

2

(3)

z

k d = 035

x 0

z

0



Acknowledgments
















z













hellip3dagger







~EE

1 2 3

r32

r 12

t 32t 21

d2

Medium-3SilverAir

d d2



















deg3 Dagger deg1





deg3 Dagger deg1


deg3 Dagger deg1



deg3 Dagger deg1













-10 00 10 20 3005

15

25

10

20

30

4020

30

40


e = -e

e = -e

e = e = -e

1 2

3 2

1 3 2




2 frac12 degk)




res ˆ para0

paraminˆ iexcl


4pd hellip15dagger











-1500 -1000 -500 00 500 1000 1500X (nm) --gt

000

005

010

015

020

025


-1500 -1000 -500 00 500 1000 1500000

010

020

030







khellip1daggerz

deg1


deg2

ˆ 0 hellip16dagger


kx ˆ

c


sup3 acute1=2

hellip17dagger










tanhkhellip2dagger

z d

2


z

khellip2daggerz

hellip18dagger

cothkhellip2dagger

z d

2


z

khellip2daggerz

hellip19dagger




p=2 and







00 20 40k k

00

05

10w

w

w = 00 w = 02w w = 2w

ll

(1)

mp

mp

mp p

p

x p

p

















00 20 40 60 80 100k k

00

02

04

06

08

10

ww

k d=03light modes

00 20 40 60 80 100k k

00

02

04

06

08

10w

w

k d=03k d=05

150 250 350 450 550 650k k

07070

07071

07072

07073

07074

100 200 300 400 500k k

07068

07069

07070

07071

07072(a) (b)

(c) (d)

x p x p

p x x p

p p

p

p

p

ll



















00 100 200 300 40010

-4

10-2

100

102

e=-1001e=-0999e=-1001+i0001

00 200 400 60010-6

10-4

10-2

100

102

104

e=-100002e=-1+i00001 e= -099998

310 315 32010-2

100

102

00 100 200 300 40010-4

10-2

100

102

104

m=-05m = 0m = +1

k d =0350

m =-11k d =035

e=-100

k d=040m=10















-2000 -1000 00 1000 2000X (nm) --gt

0000

0005

0010

0015


e=-15+04i e=-18+04i

-2000 -1000 00 1000 20000000

0010

0020


-2000 -1000 00 1000 200000

10

20














10 50 10 150k k

10-3

10-2

10-1

100

101

|T e

xp[-

(k

+k

) d

2]|

e =-15+04ie =-17+04ie =-18+04ie =-14+04i

2

(1)

2

2

2

(3)

z

k d = 035

x 0

z

0



Acknowledgments




























deg3 Dagger deg1





deg3 Dagger deg1


deg3 Dagger deg1



deg3 Dagger deg1













-10 00 10 20 3005

15

25

10

20

30

4020

30

40


e = -e

e = -e

e = e = -e

1 2

3 2

1 3 2




2 frac12 degk)




res ˆ para0

paraminˆ iexcl


4pd hellip15dagger











-1500 -1000 -500 00 500 1000 1500X (nm) --gt

000

005

010

015

020

025


-1500 -1000 -500 00 500 1000 1500000

010

020

030







khellip1daggerz

deg1


deg2

ˆ 0 hellip16dagger


kx ˆ

c


sup3 acute1=2

hellip17dagger










tanhkhellip2dagger

z d

2


z

khellip2daggerz

hellip18dagger

cothkhellip2dagger

z d

2


z

khellip2daggerz

hellip19dagger




p=2 and







00 20 40k k

00

05

10w

w

w = 00 w = 02w w = 2w

ll

(1)

mp

mp

mp p

p

x p

p

















00 20 40 60 80 100k k

00

02

04

06

08

10

ww

k d=03light modes

00 20 40 60 80 100k k

00

02

04

06

08

10w

w

k d=03k d=05

150 250 350 450 550 650k k

07070

07071

07072

07073

07074

100 200 300 400 500k k

07068

07069

07070

07071

07072(a) (b)

(c) (d)

x p x p

p x x p

p p

p

p

p

ll



















00 100 200 300 40010

-4

10-2

100

102

e=-1001e=-0999e=-1001+i0001

00 200 400 60010-6

10-4

10-2

100

102

104

e=-100002e=-1+i00001 e= -099998

310 315 32010-2

100

102

00 100 200 300 40010-4

10-2

100

102

104

m=-05m = 0m = +1

k d =0350

m =-11k d =035

e=-100

k d=040m=10















-2000 -1000 00 1000 2000X (nm) --gt

0000

0005

0010

0015


e=-15+04i e=-18+04i

-2000 -1000 00 1000 20000000

0010

0020


-2000 -1000 00 1000 200000

10

20














10 50 10 150k k

10-3

10-2

10-1

100

101

|T e

xp[-

(k

+k

) d

2]|

e =-15+04ie =-17+04ie =-18+04ie =-14+04i

2

(1)

2

2

2

(3)

z

k d = 035

x 0

z

0



Acknowledgments

















deg3 Dagger deg1





deg3 Dagger deg1


deg3 Dagger deg1



deg3 Dagger deg1













-10 00 10 20 3005

15

25

10

20

30

4020

30

40


e = -e

e = -e

e = e = -e

1 2

3 2

1 3 2




2 frac12 degk)




res ˆ para0

paraminˆ iexcl


4pd hellip15dagger











-1500 -1000 -500 00 500 1000 1500X (nm) --gt

000

005

010

015

020

025


-1500 -1000 -500 00 500 1000 1500000

010

020

030







khellip1daggerz

deg1


deg2

ˆ 0 hellip16dagger


kx ˆ

c


sup3 acute1=2

hellip17dagger










tanhkhellip2dagger

z d

2


z

khellip2daggerz

hellip18dagger

cothkhellip2dagger

z d

2


z

khellip2daggerz

hellip19dagger




p=2 and







00 20 40k k

00

05

10w

w

w = 00 w = 02w w = 2w

ll

(1)

mp

mp

mp p

p

x p

p

















00 20 40 60 80 100k k

00

02

04

06

08

10

ww

k d=03light modes

00 20 40 60 80 100k k

00

02

04

06

08

10w

w

k d=03k d=05

150 250 350 450 550 650k k

07070

07071

07072

07073

07074

100 200 300 400 500k k

07068

07069

07070

07071

07072(a) (b)

(c) (d)

x p x p

p x x p

p p

p

p

p

ll



















00 100 200 300 40010

-4

10-2

100

102

e=-1001e=-0999e=-1001+i0001

00 200 400 60010-6

10-4

10-2

100

102

104

e=-100002e=-1+i00001 e= -099998

310 315 32010-2

100

102

00 100 200 300 40010-4

10-2

100

102

104

m=-05m = 0m = +1

k d =0350

m =-11k d =035

e=-100

k d=040m=10















-2000 -1000 00 1000 2000X (nm) --gt

0000

0005

0010

0015


e=-15+04i e=-18+04i

-2000 -1000 00 1000 20000000

0010

0020


-2000 -1000 00 1000 200000

10

20














10 50 10 150k k

10-3

10-2

10-1

100

101

|T e

xp[-

(k

+k

) d

2]|

e =-15+04ie =-17+04ie =-18+04ie =-14+04i

2

(1)

2

2

2

(3)

z

k d = 035

x 0

z

0



Acknowledgments
















-10 00 10 20 3005

15

25

10

20

30

4020

30

40


e = -e

e = -e

e = e = -e

1 2

3 2

1 3 2




2 frac12 degk)




res ˆ para0

paraminˆ iexcl


4pd hellip15dagger











-1500 -1000 -500 00 500 1000 1500X (nm) --gt

000

005

010

015

020

025


-1500 -1000 -500 00 500 1000 1500000

010

020

030







khellip1daggerz

deg1


deg2

ˆ 0 hellip16dagger


kx ˆ

c


sup3 acute1=2

hellip17dagger










tanhkhellip2dagger

z d

2


z

khellip2daggerz

hellip18dagger

cothkhellip2dagger

z d

2


z

khellip2daggerz

hellip19dagger




p=2 and







00 20 40k k

00

05

10w

w

w = 00 w = 02w w = 2w

ll

(1)

mp

mp

mp p

p

x p

p

















00 20 40 60 80 100k k

00

02

04

06

08

10

ww

k d=03light modes

00 20 40 60 80 100k k

00

02

04

06

08

10w

w

k d=03k d=05

150 250 350 450 550 650k k

07070

07071

07072

07073

07074

100 200 300 400 500k k

07068

07069

07070

07071

07072(a) (b)

(c) (d)

x p x p

p x x p

p p

p

p

p

ll



















00 100 200 300 40010

-4

10-2

100

102

e=-1001e=-0999e=-1001+i0001

00 200 400 60010-6

10-4

10-2

100

102

104

e=-100002e=-1+i00001 e= -099998

310 315 32010-2

100

102

00 100 200 300 40010-4

10-2

100

102

104

m=-05m = 0m = +1

k d =0350

m =-11k d =035

e=-100

k d=040m=10















-2000 -1000 00 1000 2000X (nm) --gt

0000

0005

0010

0015


e=-15+04i e=-18+04i

-2000 -1000 00 1000 20000000

0010

0020


-2000 -1000 00 1000 200000

10

20














10 50 10 150k k

10-3

10-2

10-1

100

101

|T e

xp[-

(k

+k

) d

2]|

e =-15+04ie =-17+04ie =-18+04ie =-14+04i

2

(1)

2

2

2

(3)

z

k d = 035

x 0

z

0



Acknowledgments













2 frac12 degk)




res ˆ para0

paraminˆ iexcl


4pd hellip15dagger











-1500 -1000 -500 00 500 1000 1500X (nm) --gt

000

005

010

015

020

025


-1500 -1000 -500 00 500 1000 1500000

010

020

030







khellip1daggerz

deg1


deg2

ˆ 0 hellip16dagger


kx ˆ

c


sup3 acute1=2

hellip17dagger










tanhkhellip2dagger

z d

2


z

khellip2daggerz

hellip18dagger

cothkhellip2dagger

z d

2


z

khellip2daggerz

hellip19dagger




p=2 and







00 20 40k k

00

05

10w

w

w = 00 w = 02w w = 2w

ll

(1)

mp

mp

mp p

p

x p

p

















00 20 40 60 80 100k k

00

02

04

06

08

10

ww

k d=03light modes

00 20 40 60 80 100k k

00

02

04

06

08

10w

w

k d=03k d=05

150 250 350 450 550 650k k

07070

07071

07072

07073

07074

100 200 300 400 500k k

07068

07069

07070

07071

07072(a) (b)

(c) (d)

x p x p

p x x p

p p

p

p

p

ll



















00 100 200 300 40010

-4

10-2

100

102

e=-1001e=-0999e=-1001+i0001

00 200 400 60010-6

10-4

10-2

100

102

104

e=-100002e=-1+i00001 e= -099998

310 315 32010-2

100

102

00 100 200 300 40010-4

10-2

100

102

104

m=-05m = 0m = +1

k d =0350

m =-11k d =035

e=-100

k d=040m=10















-2000 -1000 00 1000 2000X (nm) --gt

0000

0005

0010

0015


e=-15+04i e=-18+04i

-2000 -1000 00 1000 20000000

0010

0020


-2000 -1000 00 1000 200000

10

20














10 50 10 150k k

10-3

10-2

10-1

100

101

|T e

xp[-

(k

+k

) d

2]|

e =-15+04ie =-17+04ie =-18+04ie =-14+04i

2

(1)

2

2

2

(3)

z

k d = 035

x 0

z

0



Acknowledgments
















-1500 -1000 -500 00 500 1000 1500X (nm) --gt

000

005

010

015

020

025


-1500 -1000 -500 00 500 1000 1500000

010

020

030







khellip1daggerz

deg1


deg2

ˆ 0 hellip16dagger


kx ˆ

c


sup3 acute1=2

hellip17dagger










tanhkhellip2dagger

z d

2


z

khellip2daggerz

hellip18dagger

cothkhellip2dagger

z d

2


z

khellip2daggerz

hellip19dagger




p=2 and







00 20 40k k

00

05

10w

w

w = 00 w = 02w w = 2w

ll

(1)

mp

mp

mp p

p

x p

p

















00 20 40 60 80 100k k

00

02

04

06

08

10

ww

k d=03light modes

00 20 40 60 80 100k k

00

02

04

06

08

10w

w

k d=03k d=05

150 250 350 450 550 650k k

07070

07071

07072

07073

07074

100 200 300 400 500k k

07068

07069

07070

07071

07072(a) (b)

(c) (d)

x p x p

p x x p

p p

p

p

p

ll



















00 100 200 300 40010

-4

10-2

100

102

e=-1001e=-0999e=-1001+i0001

00 200 400 60010-6

10-4

10-2

100

102

104

e=-100002e=-1+i00001 e= -099998

310 315 32010-2

100

102

00 100 200 300 40010-4

10-2

100

102

104

m=-05m = 0m = +1

k d =0350

m =-11k d =035

e=-100

k d=040m=10















-2000 -1000 00 1000 2000X (nm) --gt

0000

0005

0010

0015


e=-15+04i e=-18+04i

-2000 -1000 00 1000 20000000

0010

0020


-2000 -1000 00 1000 200000

10

20














10 50 10 150k k

10-3

10-2

10-1

100

101

|T e

xp[-

(k

+k

) d

2]|

e =-15+04ie =-17+04ie =-18+04ie =-14+04i

2

(1)

2

2

2

(3)

z

k d = 035

x 0

z

0



Acknowledgments













khellip1daggerz

deg1


deg2

ˆ 0 hellip16dagger


kx ˆ

c


sup3 acute1=2

hellip17dagger










tanhkhellip2dagger

z d

2


z

khellip2daggerz

hellip18dagger

cothkhellip2dagger

z d

2


z

khellip2daggerz

hellip19dagger




p=2 and







00 20 40k k

00

05

10w

w

w = 00 w = 02w w = 2w

ll

(1)

mp

mp

mp p

p

x p

p

















00 20 40 60 80 100k k

00

02

04

06

08

10

ww

k d=03light modes

00 20 40 60 80 100k k

00

02

04

06

08

10w

w

k d=03k d=05

150 250 350 450 550 650k k

07070

07071

07072

07073

07074

100 200 300 400 500k k

07068

07069

07070

07071

07072(a) (b)

(c) (d)

x p x p

p x x p

p p

p

p

p

ll



















00 100 200 300 40010

-4

10-2

100

102

e=-1001e=-0999e=-1001+i0001

00 200 400 60010-6

10-4

10-2

100

102

104

e=-100002e=-1+i00001 e= -099998

310 315 32010-2

100

102

00 100 200 300 40010-4

10-2

100

102

104

m=-05m = 0m = +1

k d =0350

m =-11k d =035

e=-100

k d=040m=10















-2000 -1000 00 1000 2000X (nm) --gt

0000

0005

0010

0015


e=-15+04i e=-18+04i

-2000 -1000 00 1000 20000000

0010

0020


-2000 -1000 00 1000 200000

10

20














10 50 10 150k k

10-3

10-2

10-1

100

101

|T e

xp[-

(k

+k

) d

2]|

e =-15+04ie =-17+04ie =-18+04ie =-14+04i

2

(1)

2

2

2

(3)

z

k d = 035

x 0

z

0



Acknowledgments













00 20 40k k

00

05

10w

w

w = 00 w = 02w w = 2w

ll

(1)

mp

mp

mp p

p

x p

p

















00 20 40 60 80 100k k

00

02

04

06

08

10

ww

k d=03light modes

00 20 40 60 80 100k k

00

02

04

06

08

10w

w

k d=03k d=05

150 250 350 450 550 650k k

07070

07071

07072

07073

07074

100 200 300 400 500k k

07068

07069

07070

07071

07072(a) (b)

(c) (d)

x p x p

p x x p

p p

p

p

p

ll



















00 100 200 300 40010

-4

10-2

100

102

e=-1001e=-0999e=-1001+i0001

00 200 400 60010-6

10-4

10-2

100

102

104

e=-100002e=-1+i00001 e= -099998

310 315 32010-2

100

102

00 100 200 300 40010-4

10-2

100

102

104

m=-05m = 0m = +1

k d =0350

m =-11k d =035

e=-100

k d=040m=10















-2000 -1000 00 1000 2000X (nm) --gt

0000

0005

0010

0015


e=-15+04i e=-18+04i

-2000 -1000 00 1000 20000000

0010

0020


-2000 -1000 00 1000 200000

10

20














10 50 10 150k k

10-3

10-2

10-1

100

101

|T e

xp[-

(k

+k

) d

2]|

e =-15+04ie =-17+04ie =-18+04ie =-14+04i

2

(1)

2

2

2

(3)

z

k d = 035

x 0

z

0



Acknowledgments

























00 20 40 60 80 100k k

00

02

04

06

08

10

ww

k d=03light modes

00 20 40 60 80 100k k

00

02

04

06

08

10w

w

k d=03k d=05

150 250 350 450 550 650k k

07070

07071

07072

07073

07074

100 200 300 400 500k k

07068

07069

07070

07071

07072(a) (b)

(c) (d)

x p x p

p x x p

p p

p

p

p

ll



















00 100 200 300 40010

-4

10-2

100

102

e=-1001e=-0999e=-1001+i0001

00 200 400 60010-6

10-4

10-2

100

102

104

e=-100002e=-1+i00001 e= -099998

310 315 32010-2

100

102

00 100 200 300 40010-4

10-2

100

102

104

m=-05m = 0m = +1

k d =0350

m =-11k d =035

e=-100

k d=040m=10















-2000 -1000 00 1000 2000X (nm) --gt

0000

0005

0010

0015


e=-15+04i e=-18+04i

-2000 -1000 00 1000 20000000

0010

0020


-2000 -1000 00 1000 200000

10

20














10 50 10 150k k

10-3

10-2

10-1

100

101

|T e

xp[-

(k

+k

) d

2]|

e =-15+04ie =-17+04ie =-18+04ie =-14+04i

2

(1)

2

2

2

(3)

z

k d = 035

x 0

z

0



Acknowledgments


























00 100 200 300 40010

-4

10-2

100

102

e=-1001e=-0999e=-1001+i0001

00 200 400 60010-6

10-4

10-2

100

102

104

e=-100002e=-1+i00001 e= -099998

310 315 32010-2

100

102

00 100 200 300 40010-4

10-2

100

102

104

m=-05m = 0m = +1

k d =0350

m =-11k d =035

e=-100

k d=040m=10















-2000 -1000 00 1000 2000X (nm) --gt

0000

0005

0010

0015


e=-15+04i e=-18+04i

-2000 -1000 00 1000 20000000

0010

0020


-2000 -1000 00 1000 200000

10

20














10 50 10 150k k

10-3

10-2

10-1

100

101

|T e

xp[-

(k

+k

) d

2]|

e =-15+04ie =-17+04ie =-18+04ie =-14+04i

2

(1)

2

2

2

(3)

z

k d = 035

x 0

z

0



Acknowledgments























-2000 -1000 00 1000 2000X (nm) --gt

0000

0005

0010

0015


e=-15+04i e=-18+04i

-2000 -1000 00 1000 20000000

0010

0020


-2000 -1000 00 1000 200000

10

20














10 50 10 150k k

10-3

10-2

10-1

100

101

|T e

xp[-

(k

+k

) d

2]|

e =-15+04ie =-17+04ie =-18+04ie =-14+04i

2

(1)

2

2

2

(3)

z

k d = 035

x 0

z

0



Acknowledgments























10 50 10 150k k

10-3

10-2

10-1

100

101

|T e

xp[-

(k

+k

) d

2]|

e =-15+04ie =-17+04ie =-18+04ie =-14+04i

2

(1)

2

2

2

(3)

z

k d = 035

x 0

z

0



Acknowledgments














10 50 10 150k k

10-3

10-2

10-1

100

101

|T e

xp[-

(k

+k

) d

2]|

e =-15+04ie =-17+04ie =-18+04ie =-14+04i

2

(1)

2

2

2

(3)

z

k d = 035

x 0

z

0



Acknowledgments












Acknowledgments











Documents

The asymmetric lossy near-perfect lensxlab.me.berkeley.edu/MURI/publications/publications_4.pdf · 2. The asymmetric slab Typically we consider the asymmetric lens to be a slab of