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T h e o p e n – a c c e s s j o u r n a l f o r p h y s i c s

New Journal of Physics

Near-field imaging of interacting nano objects withmetal and metamaterial superlenses

T Hakkarainen1,4, T Setala1 and A T Friberg1,2,3

1 Department of Applied Physics, Aalto University School of Science,PO Box 13500, FI-00076 Aalto, Finland2 Department of Physics and Mathematics, University of Eastern Finland,PO Box 111, FI-80101 Joensuu, Finland3 Department of Microelectronics and Applied Physics, Royal Institute ofTechnology, Electrum 229, SE-164 40 Kista, SwedenE-mail: timo.hakkarainen@aalto.fi

New Journal of Physics 14 (2012) 043019 (13pp)Received 11 January 2012Published 18 April 2012Online at http://www.njp.org/doi:10.1088/1367-2630/14/4/043019

Abstract. Employing rigorous electromagnetic theory we investigate opticalthe near-field imaging of two interacting dipole-like objects with metal andslightly lossy metamaterial nanoslab superlenses. Our analysis indicates that thedipole emission is suppressed by near-field interactions when the objects areclose to the lens or each other. This strongly influences the image quality, inparticular with objects of small size and high polarizability. The interferencefrom two nearby objects also affects the resolution and subwavelength definitioncan only be obtained for objects with dipole moments predominantly orthogonalto the slab. Such an optimal imaging condition is achieved with excitation bytotal internal reflection. With simulations we show that in these circumstances,subwavelength resolutions of about λ/5 for silver superlens and λ/10 formetamaterial slab are reached.

4 Author to whom any correspondence should be addressed.

New Journal of Physics 14 (2012) 0430191367-2630/12/043019+13$33.00 © IOP Publishing Ltd and Deutsche Physikalische Gesellschaft

2

Contents

1. Introduction 22. Imaging geometry 33. Emission properties of interacting point-like objects 4

3.1. Dipole moments for coupled emitters . . . . . . . . . . . . . . . . . . . . . . . 43.2. Influence of the dipole–dipole and dipole–slab interactions . . . . . . . . . . . 5

4. Imaging capability of metallic and metamaterial superlenses 75. Conclusions 11Acknowledgments 12References 12

1. Introduction

In conventional optical imaging, resolution is limited by the light’s wavelength λ and thenumerical aperture (NA) of the system. Optical imaging beyond the classic Rayleigh–Abbediffraction limit of λ/2 in spatial definition has for a long time attracted the attentionof scientists, and several approaches for overcoming this limit have been developed. Inbiomedical imaging, high-resolution far-field techniques, such as stimulated emission depletionmicroscopy [1], have been demonstrated. Advances in computed imaging have allowedexperimental high-definition reconstruction of three-dimensional (3D) samples; examples ofrecent methods include tomographic diffraction microscopy [2] and interferometric syntheticaperture microscopy [3] based on inverse scattering. In nanophotonics, on the other hand, anentirely different and widely employed approach is optical near-field microscopy [4], whichmakes use of (scanning) local probes and near-field imaging elements. The imaging device isplaced at a subwavelength distance from the object, enabling detection of the evanescent fieldcreated by the object and thereby increasing the effective NA. Such nanophotonic microscopyincludes evanescent-wave and multiple scattering effects, but the close proximity of the sampleand the detector also leads to near-field interactions between the device and the object. Theseinteractions may distort the image and are an inherent feature of nanoscale imaging techniques.

In near-field imaging metallic and metamaterial superlenses [5] providing subwavelengthimage resolution have inspired an increasing number of both theoretical [6–13] andexperimental [14–19] works following the suggestion of the ‘perfect lens’ by Pendry [20] in2000. The perfect lens is a lossless nanoslab of negative-index material (NIM) that wouldwork as a non-interacting imaging device delivering an undistorted image with unlimiteddefinition. However, any realistic metamaterial is necessarily lossy leading to absorptionof the object radiation and to interactions between the lens and the object. Still, theyevidently allow subwavelength resolution [10–13]. Despite the rapid progress in the field ofmetamaterials [21–28], experimental NIM-lens realizations are few [19]. Besides NIM lenses,subwavelength nanophotonic imaging of 2D objects can be achieved with physically realizablemetallic nanoslabs exhibiting plasmon resonances [14, 15, 18].

Making use of a rigorous electromagnetic theory for near-field imaging of a dipolaremitter by metallic and slightly absorbing metamaterial superlenses [11, 12], in this work weconsider such a nanophotonic imaging of two interacting point-like objects. Our analysis also

New Journal of Physics 14 (2012) 043019 (http://www.njp.org/)

3

d b

objectplane

observationplanez = 0

q1

q2

EexEim

I II III

z0

Figure 1. A slab of thickness d and material parameters εr2(ω), µr2(ω)

(medium II) is embedded between two half-space materials characterized byεr1(ω), µr1(ω) (medium I) and εr3(ω), µr3(ω) (medium III). Two polarizablepoint-like emitters are situated in the ‘object plane’ at z = z0 in front of theslab and are excited by a normally (from left) or perpendicularly (from below)incident, p-polarized plane wave Eex. The induced dipole moments q1 and q2 areinfluenced by interactions between the emitters and the nanoslab. The image-space field Eim transmitted through the slab is considered in the ‘observationplane’ at z = d + b in medium III.

accounts fully for the influence of the imaging device on the objects [13]. The object–lens andobject–object interactions lead to several remarkable and physically important effects. In closeproximity to the metal or NIM slab the dipole moment of the emitter is strongly reduced, ascan be explained in terms of a mirror-image dipole in the quasistatic limit [13]. The near-fieldinteractions between two adjacent dipoles suppress the emission of both dipoles as they comecloser together, thereby considerably attenuating the intensity of their images. We further showthat two interacting dipoles can be unambiguously resolved with subwavelength resolution onlywhen the dipole moments are predominantly oriented orthogonal to the nanoslab. Such a near-field imaging arrangement can be created through excitation by total internal reflection. Ourresults also indicate that the imaging capabilities of NIM superlenses are superior to those ofplasmonic (metallic) nanoslabs, owing to the weaker reflections and stronger enhancement ofevanescent waves by metamaterials.

2. Imaging geometry

The imaging geometry is illustrated in figure 1. Radiation from an external source is incidentinto half-space z < 0 (medium I). A transversally infinite slab of thickness d (medium II),located within 0 < z < d, works as a near-field imaging lens. The slab is followed by anotherhalf-space z > d (medium III). As the objects, we use two polarizable point-like emittersdescribed by their dipole moments q1 and q2. The emitters are located in the ‘object plane’ z =

z0 in front of the slab and the image is considered in the ‘observation plane’ a distance b behindthe slab. The objects are excited by a normally or perpendicularly incident, monochromatic,p-polarized plane wave Eex(r, ω), where ω is the angular frequency. The fields generated as

New Journal of Physics 14 (2012) 043019 (http://www.njp.org/)

4

a result of the (multiple) interactions of the incident wave with the dipole objects and theslab, including the image field Eim(r, ω) exiting into medium III, are all based on the exactboundary conditions of electromagnetic fields [13]. The time-dependent parts e−iωt of all fieldsare suppressed.

We consider two different imaging schemes [11–13]. The first is a metallic superlensstructure: a thin silver (Ag) sheet is sandwiched between two dielectric materials. Medium Iis composed of polymethyl methacrylate (PMMA), while medium III is a photoresist (PR).For a good impedance match between the materials we take the excitation wavelength asλ = 365 nm (I line of mercury lamp). At this wavelength the materials are nonmagnetic andtheir relative electric permittivities are [14, 15, 29] εr1 = 2.3, εr2 = −2.4 + i0.2 and εr3 = 2.9,respectively. As the slab thickness we use d = 35 nm, which has been demonstrated to be anappropriate value in recent experimental [14, 15] and theoretical [13] studies. To ensure thatthe contribution of the near-field components of the object radiation remains strong in theobservation plane, we consider the image intensity distribution in the PR immediately behindthe silver film (b = 2 nm). The second imaging geometry is a near-perfect lens structure in whicha slightly absorbing NIM slab is surrounded by vacuum. In this case, we choose the excitationwavelength as λ = 633 nm and the effective material parameters of the slab as εr2 = −1 + iε ′′

r2and µr2 = −1 + iµ′′

r2, where ε ′′

r2 and µ′′

r2 are small positive numbers. For the slab thickness weuse d = 35 nm, and in line with the perfect lens structure we consider the image at the distanced/2 behind the metamaterial slab.

3. Emission properties of interacting point-like objects

3.1. Dipole moments for coupled emitters

We consider interacting dipole-like objects in the vicinity of the imaging slab. The electric fieldat point r = (x, y, z) generated by a point dipole located at r0 = (x0, y0, z0) in a homogeneous,linear and isotropic medium is [13, 30]

E(r, ω) = µrµ0ω2

↔

G (r, r0, ω) · q, (1)

where µ0 is the vacuum permeability,↔

G (r, r0, ω) is the outgoing infinite-space dyadic Greenfunction and q is the dipole moment. The dipole moment of a polarizable point dipole isdetermined through a relation

q =↔

α · Etot(r0, ω), (2)

where↔

α is the polarizability and Etot(r0, ω) is the total electric field at the location of the dipole.In our geometry (see figure 1), the first dipole q1 is located at r1 = (x1, y1, z1) and the seconddipole q2 at r2 = (x2, y2, z2). We choose x2 > x1 = 0, y1 = y2 = 0, and take the dipoles at equaldistance from the slab, i.e. z1 = z2 = z0 < 0. We further assume that the polarizability of both

dipoles is isotropic, i.e.↔

α i= α↔

I , where α is a scalar quantity,↔

I is the unit tensor and i = 1, 2.The total electric field at the position of each dipole consists of the exciting field, Eex(ri , ω),the part of this field that is reflected from the slab, say Eex,r(ri , ω), and the dipole field itselfscattered back from the slab. In addition, the dipole field propagated straight from the adjacentdipole, as well as reflected from the slab, contribute to the total field. Hence, we can write

New Journal of Physics 14 (2012) 043019 (http://www.njp.org/)

5

self-consistent relations for the dipole moments

qi = α{Eex(ri , ω) + Eex,r(ri , ω) + µr1µ0ω2[

↔

Gri (ri , ri , ω) · qi +↔

G j (ri , r j , ω) · q j

+↔

Gr j (ri , r j , ω) · q j ]}, i, j ∈ {1, 2}, i 6= j, (3)

where↔

Gri (r, ri , ω) and↔

Gi (r, ri , ω) denote the reflection Green tensor and the free-space Greentensor, respectively, for the dipole located at ri . The explicit forms of the Green tensors wereworked out in detail in our earlier study [13] (see also [31, 32]). We note that (3) does notaccount for the radiation reaction of the dipole. While this effect could readily be included,amounting to a small correction [30, 33, 34] to the polarizability α, it is not of importance inour analysis since we only consider some characteristic values of α.

In the case of normal incidence, the exciting field is Eex(r, ω) = E0 eikz1z, with E0 =

(1, 0, 0) and kz1 denoting the z-component of the wave vector in medium I. For the perpendicularincidence Eex(r, ω) = E0 eikx1x , where E0 = (0, 0, 1) and kx1 is the wave vector of a planewave propagating in the +x-direction. For the normal incidence, the reflected exciting fieldat the dipole site is Eex,r(ri , ω) = −RpE0 e−ikz1z0 , where Rp is the slab reflection coefficientfor p-polarization. The minus sign originates from the choice of the unit vectors for thepolarization components [13]. For the perpendicular incidence, Eex,r(ri , ω) = 0. Noting further

that↔

Gr1 (r1, r1, ω) =↔

Gr2 (r2, r2, ω), we can solve for the dipole moments

qi = [↔

A −↔

Bi

↔

A−1↔

B j ]−1[bi +

↔

Bi

↔

A−1

b j ], i, j ∈ {1, 2}, i 6= j, (4)

where↔

A=↔

I −αµr1µ0ω2

↔

Gri (ri , ri , ω), i = 1, 2, (5a)

↔

Bi= αµr1µ0ω2[

↔

G j (ri , r j , ω)+↔

Gr j (ri , r j , ω)], i, j ∈ {1, 2}, i 6= j, (5b)

bi = α[Eex(ri , ω) + Eex,r(ri , ω)], i = 1, 2. (5c)

For the normally incident excitation, Eex(r1, ω) = Eex(r2, ω) and Eex,r(r1, ω) = Eex,r(r2, ω),implying that b1 = b2. In contrast, for the perpendicular excitation Eex(r1, ω) and Eex(r2, ω)

differ from each other by a phase factor. One may also notice that↔

G1 (r2, r1, ω) =↔

G2 (r1, r2, ω) in the case of both excitations.

3.2. Influence of the dipole–dipole and dipole–slab interactions

Using (4) we now investigate the dipole–dipole and dipole–slab interactions. We calculate thedipole moments of the point emitters as a function of their distance from the slab, as wellas their distance from each other. For α we use the values 5 × 10−33 and 5 × 10−35 cm2 V−1

that, according to the Clausius–Mossotti relation [35], characterize the polarizabilities of ametallic sphere of radius ∼20 nm and a glass sphere of radius ∼10 nm, respectively, in vacuumor PMMA. We also study the interactions with molecule-like objects, for which we takeα = 1 × 10−30 cm2 V−1, consistent with atomic dipole moments and unit-amplitude incidentelectric field. We focus on the near-field effects, so the distances of the emitters from theslab and from each other are within the ranges 10 nm6 |z0|. λ1/2 and 10 nm61x . λ1/2,

New Journal of Physics 14 (2012) 043019 (http://www.njp.org/)

6

0 100

200-200

-1000

5

10

15

Δx (nm)

|q

| (C

m)

x1

x 10-33

(nm)

5

10

15

5

10

5

10

0 100

200-200

-1000

|q

| (C

m)

z1

x 10-33

Δx (nm) (nm)

0 100

200-200

-1000

|q

| (C

m)

z1

x 10-33

Δx (nm) (nm)0 100

200-200

-1000

(a) (b)

(d)(c)Δx (nm)

|q

| (C

m)

x1

x 10-33

(nm)

z0 z0

z0 z0

Figure 2. Magnitudes of the dipole moment components as a function of thedipole–slab distance z0 and the separation 1x of the dipoles. The imagingstructure is (a, b) a silver slab surrounded by PMMA and PR (λ = 365 nm,εr1 = 2.3, εr2 = −2.4 + i0.2, εr3 = 2.9) and (c, d) an absorbing metamaterial layerin vacuum environment (λ = 633 nm, εr2 = µr2 = −1 + i0.1). The objects areexcited by (a, c) a normally and (b, d) a perpendicularly incident p-polarizedplane wave. The other parameters are d = 35 nm and α = 5 × 10−33 Cm2 V−1 inall cases.

respectively, where 1x = x2 − x1 and λ1 is the excitation wavelength in medium I. Figures 2(a)and (b) illustrate the magnitudes of the dipole moment components in the case of the silver slabstructure. Figure 2(a) shows |qx1| with normal illumination (|qx2| = |qx1|), and figure 2(b) showsthe value of |qz1| for the perpendicular excitation (|qz2| ≈ |qz1|). Likewise, figures 2(c) and (d)contain the magnitudes of the dipole moment components when a slightly lossy NIM slab isused as the imaging element (εr2 = µr2 = −1 + i 0.1). These parameters correspond to a figureof merit (FOM) equal to 10, which is close to the FOM = 8.5 of a recently proposed low-lossNIM structure working at 650 nm [28]. Figure 2 (c) depicts the value of |qx1| for the normallyincident excitation, whereas figure 2(d) shows |qz1| with perpendicular illumination.

The dipole–slab near-field interaction manifests itself in figure 2 as the decrease of |qxi |

and |qzi | (i = 1, 2) when the objects approach the silver or NIM superlens. In the case of thesilver slab structure and normal excitation (see figure 2(a)) |qx1| (and |qx2|) shows an oscillatorybehavior as a function of z0. This phenomenon results from the interference of the exciting fieldwith the backscattered part of the exciting field. In contrast, due to the good impedance matchof the NIM superlens the reflections from the slab in that case are weak, making |qx1| (and |qx2|)approach quickly the constant value of α cm2 V−1 when |z0| increases (see figure 2(c)). Regularoscillations as a function of z0 are also absent when the objects are illuminated perpendicularly

New Journal of Physics 14 (2012) 043019 (http://www.njp.org/)

7

(see figures 2(b) and (d)), because Eex,r(ri , ω) = 0. All these effects are consistent with (anddiscussed more deeply in) our earlier study, in which we considered the interaction of a singledipolar object with imaging devices of this type [13].

The interaction of the objects with each other is also visible in figure 2. In all these graphsone can note an interesting (and unexpected) phenomenon: if two point-like emitters are placedvery close to each other, their dipole moments die out. This effect happens even when thedipoles are located so far from the slab that the backscattered near-field components of thedipole radiation are insignificant. Thus the effect is only a consequence of the straight near-field interaction of the adjacent dipoles. Although not explicitly shown here, this result canbe derived analytically from (4). The same phenomenon appears, at least indirectly, also inearlier studies [34, 36, 37], but the vanishing of the dipole moments is conventionally avoidedby considering small spheres which cannot be closer to each other than 2Rs, where Rs is theradius of the spheres. The peaks and ridges in 2(b) and (d) arise from the configurational

resonances [38] of the system, i.e. when det(↔

A −↔

Bi

↔

A−1↔

B j) ≈ 0 (see (4)). The resonancesare narrower and more pronounced in conditions where the quasistatic approximation holds;otherwise (as in figures 2(a) and (c)) they are smeared out.

When metallic nanospheres are near the slab (|z0|. 60 nm) and close to each other (|1x |.60 nm), the near field radiated by the adjacent emitter and reflected from the slab contributes tothe emission of the objects. With normal illumination this interaction induces small qz1 and qz2

components. Similarly, small qx1 and qx2 components for the emitters emerge when the objectsare excited perpendicularly. In the NIM-slab case these effects are negligible, but for the morereflective silver slab the minor dipole moment components show a clear influence on the electricfields of the emitters. The object–object and the object–slab interactions depend strongly on thepolarizability of the objects. For α = 5 × 10−35 cm2 V−1 (small glass sphere) it is found thatthe interactions effectively vanish. With α = 1 × 10−30 cm2 V−1 (characteristic of a molecule)the interactions, in turn, are stronger than with α = 5 × 10−33 cm2 V−1 (small metal sphere) andextend over the whole near-field range.

By virtue of the above discussions we can conclude that unless the two metallic or glassnano objects are in very close proximity to each other (i.e. 1x < 3Rs) or to a silver or NIMslab (i.e. |z0| < 3Rs), the object–object and object–slab near-field interactions do not stronglyinfluence the emission properties of the objects. However, if two molecules of high polarizabilityare brought to within near-field distances from each other (1x < λ1/2) or the imaging slab(|z0| < λ1/2), then the effects of the interactions within dipole approximation, such as thesuppression of emission, are strong.

4. Imaging capability of metallic and metamaterial superlenses

Next we analyze the imaging capabilities of both superlens structures in terms of imageresolution. The field of each dipole at the observation region in half-space z > d is

Edi,t(r, ω) = µr1µ0ω2

↔

Gt (r, ri , ω) · qi , i = 1, 2, (6)

where↔

Gt (r, ri , ω) is the Green tensor for transmission across the slab [13] and qi can becalculated from (4). A part of the exciting field is also transmitted through the slab when theobjects are illuminated by a normally incident wave. Hence, the total field in the observation

New Journal of Physics 14 (2012) 043019 (http://www.njp.org/)

8

plane is

Eim(r, ω) = Ed1,t(r, ω) + Ed2,t(r, ω) + Tp E0eikz3z, (7)

where Tp is the slab’s transmission coefficient for p-polarization [13] and kz3 denotes thez-component of the wave vector in medium III. In the case of perpendicular illumination thelast term in (7) is absent.

Owing to the dominant qx -components, the fields generated by normally excited objectsclosely resemble those created by purely x-oriented point dipoles. Consequently, the near-field intensity distribution in the observation plane (x, y) is elongated along the x-axis andcontains two peaks with a dent in the middle [11, 12]. On the other hand, with perpendicularexcitation the near-field image intensity profiles are circularly symmetric due to the majorqz-components [11, 12]. When the emitters are close to each other and close to the silver slab,the qz-components with normal illumination and qx -components for the perpendicular excitationalso contribute to the emission, as discussed above. The total near-field image-intensitydistribution in the observation plane, |Eim(r, ω)|2, contains the interference of two correlatedpoint-like objects. Figure 3 shows the image profiles along the x-axis in the observation planewhen a metallic ((a) and (b)) or a slightly lossy metamaterial ((c) and (d)) slab is used as theimaging element (see figure 1). Figures 3(a) and (c) correspond to normal excitation (x-orienteddipoles), while figures 3(b) and (d) represent perpendicular illumination (z-oriented dipoles).The system parameters are given in the figure caption.

Figures 3(a) and (c) show that the interference of two normally excited point-likeobjects makes the image intensity profiles obscure with multiple maxima. From the intensitydistributions one can recognize two point emitters if they are separated by a distance of λ1 (blackdotted lines), but already when 1x = λ1/2 it is hardly possible to identify two objects from theimage (blue dashed-dotted lines). The simulations show that unlike the case of uncorrelateddipoles [11, 12], subwavelength near-field imaging of two correlated point-like emitters is notpossible if the objects are excited in such a way that their dipole moments are predominantlyparallel to the surface of the imaging element. However, when the objects are illuminated bya perpendicularly incident plane wave (see figures 3(b) and (d)), the interference does notdisturb the imaging as much: two clear intensity peaks form in the observation plane and theirpositions match with the object locations in the object plane. In this case we are able to definethe resolution of the image as follows: two-point objects are resolved if the central dent in the|Eim(r, ω)|2 distribution is less than 81% of the lower maximum of the profile (the Rayleighcriterion). According to this criterion we find here resolutions of ∼λ1/5 and ∼λ1/10 for themetallic and metamaterial superlens structures, respectively (green solid lines in figures 3(b)and (d)). The corresponding 3D intensity distributions in the observation plane are shown infigure 4.

The image definition obtained for the silver superlens is close to the experimentally reachedresolutions of λ/6 and λ/7 for the same kind of imaging systems but for 2D objects [14, 15].Recently, a resolution of λ/12 for a 2D grating object was achieved using an ultra-thin andsmooth silver superlens [18]. However, these experiments are not directly comparable with ourimaging system: the dipole emission is three dimensional containing both s- and p-polarizationcomponents, whereas in the experiments with 2D objects only p-polarized light is present. Thesilver slab enhances the p-polarized evanescent waves due to the surface plasmons. The betterresolution for the metamaterial superlens originates from the lower losses in NIM and the greaterenhancement of evanescent wave components of the object radiation in transmission across the

New Journal of Physics 14 (2012) 043019 (http://www.njp.org/)

9

−120 0 120 2400

2

4

x (nm)360

(a)

6 Δx = λ / 4 (60 nm) 1

1Δx = λ / 2 (120 nm) Δx = λ (240 nm) 1

−100 0 100 2000

2

4

6

x (nm)300

(d)

4000 2000

1

2

3

x (nm)600

(c)

400

−120 0 120 2400

1

2

3

x (nm)(b)

|E

|2im

|E

|2im

|E

|2im

|E

|2im

Δx = λ / 4 (60 nm) 1Δx = λ / 2 (120 nm) 1

Δx = λ / 5 (50 nm) 1

Δx = λ / 4 (150 nm) 1

1Δx = λ / 2 (300 nm) Δx = λ (600 nm) 1

Δx = λ / 4 (150 nm) 1Δx = λ / 2 (300 nm) 1

Δx = λ / 10 (60 nm) 1

Figure 3. Image-intensity distributions along the x-axis in the observationplane for (a, b) a PMMA-Ag-PR superlens (λ = 365 nm, d = 35 nm, b = 2 nm,εr1 = 2.3, εr2 = −2.4 + i0.2, εr3 = 2.9) and (c, d) an absorbing NIM lens invacuum (λ = 633 nm, d = 35 nm, b = d/2, εr2 = µr2 = −1 + i0.1). The objectsare excited by (a, c) a normally or (b, d) a perpendicularly incident planewave and located at r1 = (0, 0, −40 nm) and r2 = (x2, 0, −40 nm), with λ1/106x2 6 λ1. The polarizability of the objects is α = 5 × 10−33 cm2 V−1 (typical of ametallic sphere with a radius of ∼20 nm).

slab. Furthermore, in the case of the NIM lens, both the p-polarized and the s-polarized near-field components are enhanced [11–13].

We also calculated the resolution of the imaging systems when the point emitters hadthe polarizability characterizing a molecule, but did not find any significant difference inresolution compared to the metallic nano objects. However, the brightness of the imageswas approximately 10–20 times higher for molecular-like emitters, which is a result of twocompeting phenomena: the larger polarizability increases the strength of the dipole but alsoleads to stronger near-field interactions, decreasing the emission. In addition, we analyzed theimaging of metal nanospheres and molecule-like objects with a silver superlens having slabthicknesses of 25 and 15 nm [18]. However, there was no remarkable improvement in theresolution, and the only benefit was an increase in image brightness. Furthermore, we verifiedwith simulations that lowering the absorption of NIM leads to better resolution and higher image

New Journal of Physics 14 (2012) 043019 (http://www.njp.org/)

10

(a)

-80-400 40

80

-80-40040801200

1

2

3

x (nm)

|E

|2im

y (nm)

(b)

-80-400 40

80

-80-40040801200

1

2

3

x (nm)y (nm)

4

5

|E

|2im

Figure 4. Image-intensity profiles in the observation plane. Two point-likeobjects are separated (a) by λ1/5 (r1 = (0, 0, −40 nm), r2 = (50 nm, 0, −40 nm))and the imaging element is a PMMA-Ag-PR superlens, and (b) by λ1/10 (r1 =

(0, 0, −40 nm), r2 = (60 nm, 0, −40 nm)), imaged with a slightly absorbingNIM lens. The objects are excited by a perpendicularly (from below) incidentplane wave. Other system parameters are the same as in figure 3.

θc

I

II

n p

Eex

Einc

Figure 5. Schematic illustration of excitation with a prism. A p-polarized waveis incident on the interface between the prism and medium I (np > n1) at an anglethat is slightly larger than the critical angle. The exciting field in medium I then isan evanescent wave traveling along the x-axis and decaying in the z-direction. Itselectric field component is effectively perpendicular to the surface of the imagingelement (medium II).

brightness because the evanescent wave components of the object radiation are enhanced muchmore and in a wider spatial-frequency range in the transmission through the slab [11].

On the basis of the above results we conclude that near-field imaging beyond theconventional resolution limit of λ/2 is achievable with the silver and absorbing NIM superlensstructures. To obtain subwavelength definition the excitation has to be such that the dipolesoscillate predominantly orthogonal to the imaging element. This situation can be arranged, forinstance, by placing a prism of refractive index np near the objects (see figure 5). In this setup, ap-polarized wave Einc is incident on the surface between the prism and medium I containing theobjects (np > n1) with an angle of incidence θ larger than the critical angle θc. Consequently,the exciting field in medium I is an evanescent wave traveling along the interface (x-direction)

New Journal of Physics 14 (2012) 043019 (http://www.njp.org/)

11

and having a z-dependence [39]

Eex(z) = Einc2 cos θ e−z/dp

n2 cos θ + i(sin2 θ − n2)1/2[−i(sin2 θ − n2)1/2ux + sin θ uz], (8)

where n = n1/np, and ux , uz are the Cartesian unit vectors. The parameter dp denotes thepenetration depth [39]

dp =λ

2π(n2p sin2 θ − n2

1)1/2

, (9)

which corresponds to the distance at which the evanescent wave has decreased by a factor of1/e from its value at the interface. One can readily see from (8) that when θ is only slightlylarger than θc, the exciting evanescent wave has an electric field substantially in the z-direction.With an appropriate choice of prism material this wave can be made to nearly vanish beforethe surface of the imaging slab. Consequently, the reflected part of the exciting field is zero astaken in our simulations with perpendicular illumination. As an example, in the silver superlenscase, by choosing np = 1.7 (n1 = 1.52 for PMMA) we have θc = 63

◦

. Then, using θ = 66◦

withλ = 365 nm the penetration depth takes on the value 170 nm, which is suitable for our systemwhere the objects are located 40 nm from the lens. With these parameters the z-component of theexciting field is five times as large as the component in the x-direction. This kind of excitationcorresponds well to the situation, analyzed by simulations above, in which the object dipolesare aligned orthogonal to the imaging slab.

5. Conclusions

In summary, we have studied the near-field imaging of two interacting nanoscale objects withmetallic and metamaterial superlenses using exact electromagnetic theory. Our main aim was todetermine how the near-field interactions of the objects with each other and with the imagingelement affect their emission properties and the image resolution. These issues play a significantrole in nanophotonic near-field imaging applications.

We found that the near-field interactions in the dipole model suppress the emission ofthe objects when the emitters approach the imaging slab. Likewise, the dipole moments ofthe emitters vanish when the objects are in very close proximity to each other. At certaindistances from each other and from the slab, the dipole moments exhibit a peak-like behavioras a result of configurational resonances. The effects that arise from the object–slab interactionsare weaker and do not reach as far with the NIM slab structure as they do with the metallicimaging element, a consequence of the good impedance match between the slightly lossy NIMand its surroundings. The observed phenomena depend on the polarizability of the objects: formolecular-like emitters of high polarizability, the influence of the interactions on the objectemission extends over the entire near-field range, whereas for metallic or glass nanospheres oflower polarizability, the interactions are insignificant unless the objects are nearly in contactwith each other or the imaging slab.

We showed that both lens systems can provide image resolutions beyond the classicdiffraction limit of λ/2. However, the interference of the fields from two correlated point-likeemitters affects the resolution in an essential way. Subwavelength definition cannot be achievedif the object dipoles oscillate mostly parallel to the imaging element. In contrast, if the emittersare excited such that their dipole moments are orthogonal to the slab surface, we obtain the

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resolutions of about λ/5 and λ/10 for the metallic and slightly lossy metamaterial superlensstructures, respectively.

Acknowledgments

This work was partly funded by the Academy of Finland (project 128331). TH thanks theFinnish Graduate School of Modern Optics and Photonics, the Emil Aaltonen Foundation andthe Finnish Foundation for Technology Promotion for financial support.

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