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Subwavelength imaging with arrays of plasmonic scatterers
Optics Communications 285 (2012) 3363–3367
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Optics Communications journal homepage: www.elsevier.com/locate/optcom
Subwavelength imaging with arrays of plasmonic scatterers Stanislav I. Maslovski Departamento de Engenharia Electrotécnica, Instituto de Telecomunicações, Universidade de Coimbra, Pólo II, 3030-290 Coimbra, Portugal
a r t i c l e
i n f o
Article history: Received 7 November 2011 Received in revised form 20 December 2011 Accepted 21 December 2011 Available online 3 January 2012
a b s t r a c t Here, I briefly review the key principles of subwavelength imaging in dense arrays of resonant scatterers, with an emphasis on resonant plasmonic structures at optical frequencies. © 2012 Elsevier B.V. All rights reserved.
Keywords: Subwavelength imaging Plasmonic resonance Dipolar arrays
1. Introduction The resolving power of conventional optical instruments such as microscopes is limited by the Abbe diffraction limit [1]. The nature of this limit becomes evident in the Fourier optics when the optical field is decomposed into components with different spatial frequencies, that is, with different values of the transverse wavenumber kt. Conventional lenses act as filters on this spatial spectrum, which select only the components within a limited range of wavenumbers: 0 ≤ kt ≤ kt, max b 2π/λ. Therefore, the spatial resolution of such devices is limited by δ0 ≈ π/kt, max. With the advent of metamaterials [2] it became possible to widen the spectrum of spatial frequencies to which an optical instrument may react and in this way to increase the resolving power. Namely, if optical resolution better than a half wavelength is required, it is necessary for an optical device to react to the components of the source field characterized by the wavenumbers kt > 2π/λ. These waves exist only in the near field of a source, as they decay exponentially away from the source. While such evanescent components may be picked up with a closely located sensor of the characteristic dimension on the order of the required resolution, it may be as well done with metamaterial slabs or metasurfaces that support oscillatory states localized on the scale of the required resolution. The well-known theoretical concept of perfect lensing [3] with a planar slab of a double negative metamaterial characterized by εr = − 1 and μr = − 1 (at a certain frequency of operation) belongs to the latter category, because the free surface of such ideal Veselago medium [4] supports surface plasmon-polaritons with arbitrary wavenumbers kt [5,6], and, thus, localized oscillations of arbitrarily small spatial scale can be excited on this surface. As discussed ahead, a similar behavior can be realized without employing any volumetric metamaterials. For example, one may obtain an analogous effect with sheets of nonlinear media that realize the phase 0030-4018/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2011.12.079
conjugation operation [7–10] or, under certain limitations, with arrays of weakly interacting and linear resonant inclusions (which may be plasmonic particles in optics or metallic inclusions at microwaves) [11–19]. In what follows, I review the principles of operation of such linear metasurfaces capable of picking up and restoring the evanescent components of the source field. 2. Subwavelength imaging in arrays of resonant inclusions It is convenient to start with identifying the general requirements on the transfer function of an optical system that is capable of resolving details much finer than a half wavelength. The transfer function T(ω, kt) can be formally introduced as a coefficient of proportionality between the field components of interest at the source plane and at the image plane of an imaging system. With this definition at hand, the time-harmonic optical electric field (here I consider a fixed polarization and work with scalar field) in the image plane can be written as 2
Eðω; rt Þ ¼ ∫∫
d kt ik ·r 2 ′ src −ik ·r′ T ðω; kt Þe t t ∫∫d r t E ðω; rt Þet t ; ð2πÞ2
ð1Þ
where Esrc ðω; rt Þ is the field at the source plane, with rt and r′t being the radius vectors in the image and the source planes, respectively. I use the time dependence of the form exp(− iωt). Evidently, Eðω; rt Þ ¼ Esrc ðω; rt Þ when T ðω; kt Þ≡1. This is the case of the perfect lens that restores all spectral components of the source field at the image plane, among which there are the propagating waves with jkt j b 2π=λ, and the evanescent waves with jkt j > 2π=λ. Typically, the source is placed at a certain distance from the frontal surface of the lens. There is as well a finite separation from the back side of the lens to the focal plane. Therefore, the total transfer
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S.I. Maslovski / Optics Communications 285 (2012) 3363–3367
Fig. 1. Left: A pair of two-dimensional arrays of resonant particles (plasmonic spheres) separated by distance d, operating as a subwavelength imaging device. Right: Typical dispersion curve for a proper surface mode on an array of plasmonic spheres which is required for subwavelength imaging at the frequency ω0.
function T ðω; kt Þ is, generally, a product of three components that take into account the propagation in the two gaps before and after the lens, and the propagation in the lens itself. In order for the above interpretation to be rigorous, the scalar transfer functions must be replaced with transfer matrices of the air gaps and the lens [11]. Then multiplying these matrices sequentially produces the total transfer matrix. Assuming the same thickness d/2 of the two gaps, one may write for the total transfer matrix ikjj d
T ðω; kt Þ ¼
e
2
! 0ik
ikjj d
·T 0 ðω; kt Þ· e
2
! 0ik
; ð2Þ jj d jj d 0 e− 2 0 e− 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where kjj ¼ ðω=cÞ2 −k2t is the propagation factor along the optical axis of the imaging system. The matrix T 0 ðω; kt Þ represents the transfer matrix of the lens. Evidently, for the complete device to operate as a perfect lens this matrix must be the inverse of the product of the two matrices of the air gaps. In this case the total transfer matrix of the system becomes an identity matrix. As is known, a planar slab of ideal Veselago medium [3,4] with εr = μr = − 1 and the thickness d has the required property, as it essentially negates the effect of wave propagation in a free space layer of the same thickness, and, thus, can be used as the middle component in Eq. (2). However, there is a possibility to obtain a similar effect without employing any volumetric metamaterials. One may notice that the important physical phenomena, namely, the negative refraction of the incident propagating waves and the resonant excitation of coupled surface plasmon-polaritons under the incidence of the evanescent waves happen at the two surfaces of the Veselago slab. Therefore, one might wish to examine a system composed of two metasurfaces separated with just air and look for the characteristics of these surfaces such that the total system would resemble a perfect lens. In this way we found in [7] that a pair of surfaces that phase-conjugate both electric and magnetic fields in air has the required properties of a perfect lens. Later other authors proposed devices based on similar principles [8–10]. The phase conjugation (time reversal) operation requires nonlinearities and wave mixing which I do not consider here. However, if one is interested only in the evanescent part of the spectrum, the necessary properties may be obtained without employing any nonlinearities. One may start with the transfer matrix of the lens in the form [11] T 0 ðω; kt Þ ¼
a~ c~
b~ ⋅ eikjj d ~ d 0
0
−ikjj d
e
e~ ⋅ g~
f~ ; ~ h
ð3Þ
where the two matrices on the left and on the right with yet unknown components represent the metasurfaces. What must be the components of these matrices so that the transfer matrix T 0 ðω; kt Þ becomes the inverse of the transfer matrix of an air layer, and, thus, the imaging system acquires the perfect lens
property? One may verify that this happens when a~ ¼ d~ ¼ 0, e~ ¼ h~ ¼ 0, and b~ g~ ¼ c~f~ ¼ 1. Quite interestingly, such an abstract mathematical result that follows just from the rules of matrix multiplication is actually physically realizable with dense two-dimensional arrays of resonant inclusions when they are excited by evanescent waves (see Fig. 1). By analogy with the operation of Pendry's lens [3], subwavelength imaging in such arrays occurs when they support surface modes (surface plasmon-polaritons) with the propagation factors matching the transverse wavenumbers of the incident evanescent waves. As explained in Appendix A, when a pair of arrays separated by distance d is under incidence of an evanescent wave with the frequency ω and the transverse wave vector kt that satisfy the dispersion relation for the surface modes on the arrays α
−1
ðωÞ−βðω; kt Þ ¼ 0;
ð4Þ
where α(ω) is the inclusion polarizability and βðω; kt Þ is the interaction constant of the array, the transmission coefficient through the array pair equals s21 ðω; kt Þ ¼ −exp −ikjj d for this wave. Thus, the evanescent components are amplified when tunneled through the arrays, so that the decay of the same waves in the regions before and after the arrays gets compensated. This effect was theoretically explained and experimentally verified for arrays of electrically polarizable inclusions at microwaves in our initial works [11,13] and, for magnetically polarizable inclusions, in [20]. One may notice that at a given frequency ω the dispersion relation Eq. (4) is satisfied only at a certain kt. Nevertheless, when the dispersion of the surface waves is enough flat in kt (such as shown in Fig. 1, right), practical examples show that the amplification of the evanescent waves may happen in a wide range of wavenumbers that is enough for subwavelength imaging at a fixed frequency. 3. Imaging in arrays of plasmonic particles and related challenges The realizations based on arrays with both electric and magnetic response considered in [11] (see also Appendix A) are idealized structures that aim at both the subwavelength resolution and the matching (that is, the absence of reflection). When the level of reflection is not important, one may employ the particles with only electric response. At optical and near-infrared frequencies a natural choice for such inclusions may be nanoparticles made of noble metals. In the optical band such metals exhibit plasmonic behavior due to the material response which is typically described by the Drude dispersion [22]: εr ¼ ε∞ −
ω2p ; ωðω þ iΓ Þ
ð5Þ
where ωp is the plasma frequency of the electrons in the metal, and Γ is the collision frequency that represents the loss. More precise models include additional Lorentz-type dispersion terms.
S.I. Maslovski / Optics Communications 285 (2012) 3363–3367
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The resonant frequency of a small spherical nanoparticle embedded into a dielectric with the relative permittivity εh may be estimated from the relation for the quasi-static polarizability of such a particle: 3
α ¼ 4πεh R
εr −εh ; εr þ 2ε h
ð6Þ
where R is the radius of the nanoparticle. Neglecting the effect of loss, the polarizability resonates at the frequency ω0 where εr(ω0) ≈ − 2εh. For more precise formulation one may use the well-known Mie theory. Over the last decade, subwavelength imaging with arrays of such nanoparticles was successfully demonstrated in a number of works [12,14–19]. In order to identify which structures are more promising for the subwavelength imaging purposes, one must study the surface waves supported by the structures. For instance, in [19] the dispersion of surface modes of several types and polarizations on a rectangular array of plasmonic spheres was considered. Some search for structures with optimal particle polarizabilities and flat dispersion of the surface waves was performed in [15,18]. On the other hand, in [17] the frequency scanning-based subwavelength imaging was proposed, for the plasmonic arrays with profound dispersion. In most of the mentioned works the nanoparticles are typically made of silver, that has the loss tangent |εr″/ε′r| ≈ 10 − 2 … 10 − 1 in the optical and near infrared bands [22]. For the distance between the arrays on the order of a half wavelength, this amount of loss limits the resolution by δ0 ≈ λ/5 … λ/4 (at terahertz or microwave frequencies the loss in metal particles is much lower and the resolution is limited by the period of the arrays mostly). The resolution can be improved in more closely positioned arrays, or in arrays embedded into an active host that compensates the loss. Aperiodicity in the positioning of the particles in the arrays may affect the resolution in the same manner as loss, due to random scattering. This effect has not yet been thoroughly studied. The finite size of the arrays has an insignificant influence on the subwavelength imaging performance when compared to the effect of loss. For instance, in [14] the arrays of 20 × 20 plasmonic spheres have been studied. While a considerable difference in the optimal operational frequency of the lens was observed (which is related to a different effective value of the interaction constant βðω; kt Þ in a finite array when compared to an infinite array), the obtained results generally follow the predictions of the theory initially developed for the infinite arrays. The examples available from the literature show that the surface modes polarized orthogonally to the plane of the array have a dispersion with less pronounced dependence on kt than the modes with the nanoparticles polarized tangentially to the array plane, and, thus, may resonate with the incident evanescent waves in a broader range of wavenumbers. Nevertheless, in a relatively recent paper [18], using the particles of oblate spheroidal shape a resolution on the order of λ/5 (with the separation between the image and the source plane not less than a half wavelength) was demonstrated for both main polarizations. Although in this paper I concentrate on imaging with arrays of plasmonic scatterers, the same principles apply to complementary structures where the particles are replaced with holes in a sheet of plasmonic metal. For example, in [21], Huang et al. proposed a nearfield lens formed by an array of fractal-shaped holes in a metal plate. Completely analogous to the cases considered above, the authors of [21] aimed for a flat dispersion of the surface modes (surface plasmon-polaritons) of both main polarizations, which was optimal for the subwavelength imaging. They reported such flat bands of a structured silver film at the wavelengths of about 3750 nm (80 THz) and 7500 nm (40 THz), at which |ε′r| of silver is on the order of 10 3, that is, it already approaches the properties of a perfect electric conductor. Therefore, the resolution in this type of lens is limited mostly by the structural period (which is on the order of λ/7 in their case), as in the early microwave experiments [11,13,20].
Fig. 2. The unit cell of a double array of plasmonic particles, as simulated in CST Microwave Studio™. The structure is periodic along the x and y axes with the period a.
However, at optical wavelengths the situation is significantly different: typically, one has to reduce the amount of metal in a structure to obtain a higher quality factor and better resolution. In the next section I consider such a situation where the silver nanoparticles are “shrunk” in the directions parallel to the array plane, which in the end may allow for a resolution on the order of the same λ0/7, but in the visible. 4. Numerical example: imaging in arrays of prolate spheroidal plasmonic nanoparticles Here, I consider subwavelength imaging in coupled arrays of plasmonic prolate spheroids. The geometry of the spheroids is shown in Fig. 2. The spheroids are elongated along the axis orthogonal to the array plane (the z-axis). As discussed in the previous section, such orientation allows for a flat dispersion with respect to the transverse component of the wave vector. The prolate spheroidal plasmonic inclusions may also have a significantly lower resonant frequency. Namely, in the quasi-static approximation the polarizability of such particles along the longest axis reads ε h ðεr −εh Þ ; εh þ Nðεr −εh Þ
α¼V
ð7Þ
where V is the inclusion volume and N is the depolarization factor N¼
1 −1 e2
1 1þe ln −1 ; 2e 1−e
ð8Þ
pffiffiffiffiffiffiffiffiffiffiffiffi where e ¼ 1−r 2 is the eccentricity that is calculated from the equatorial-to-polar axis ratio r of the ellipsoid. For prolate spheroids with r = 1/3, N ≈ 1/10, and, therefore, the plasmonic resonance of an isolated nanoparticle occurs at the frequency where εr ðω0 Þ≈− 1−N N ε h ≈−9ε h . From here, for silver nanoparticles with ωp ≈ 2π × 2.18 × 103 THz (Ref. [22]) embedded into a host with εh = 2.12 (e.g., silicon dioxide), one obtains from Eq. (5) that the plasmonic resonance red-shifts in this case from the near ultraviolet to the visible and occurs at the vacuum wavelength of about 6.5 × 10 2 nm. When the nanoparticles are assembled into arrays, the resonance may experience a further shift, because of the particle interactions. Namely, full-wave simulations of the structure depicted in Fig. 2 demonstrate that in arrays of prolate spheroidal silver nanoparticles with r = 1/3, R = 2a/3 and the array period a = λh/10 with λh ¼ pffiffiffiffiffi λ0 = εh and εh = 2.12, the plasmonic resonance occurs at around λ0 = 660 nm. At this wavelength (hν = 1.88 eV) the complex permittivity of silver is εr′ ¼ −20 0:1 and εr″ ¼ 0:45 0:18 [23], thus, the loss tangent of the material of the nanoparticles is within 0.013 b tan
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S.I. Maslovski / Optics Communications 285 (2012) 3363–3367
Fig. 3. The complex transfer function T(ω0, ky) of a pair of arrays of plasmonic silver particles with the period a = 45.3 nm separated by the distance d = 2.5a. The absolute value of T (on the left) and the argument of T (on the right) are plotted as functions of the normalized transverse wave vector component kya, for the indicated values of the loss tangent of the plasmonic metal. The array particles are prolate spheroids with the polar radius R = 2a/3 and the equatorial radius of R/3. The particles are embedded in a dielectric with permittivity εh = 2.12. The wavelength of operation (in vacuum) is λ0 = 660 nm.
δ b 0.032. The loss in the host dielectric is typically much lower and may be neglected. In order to demonstrate the influence of loss on the subwavelength imaging in arrays of silver nanoparticles, I solve with full-wave simulations for the total complex transfer function (including the air gaps) T(ω0, ky) of a pair of such arrays separated by d = λh/4. These results are shown in Fig. 3 for the case of P-polarized incident light with the transverse wave vector kt oriented along the y-axis. Assuming that the loss is at the lower limit, I obtain a transfer function that over amplifies the evanescent components with kya ≈ 1.2 (the dashed blue curve). On the contrary, when the loss is at the upper limit, one may observe a significant dumping of the components with kya > 1.4 as compared to the case of low loss (the solid green curve). One may also notice a dip on these plots at kya = 2πa/λh, which is at the boundary between the propagating and the evanescent part of the spatial spectrum. In both considered cases, the transmitted spectrum is practically completely cut off at around kya = 2. Nevertheless, besides the mentioned non-idealities in the transfer function, such a near-field lens is actually able to produce rather sharp images of point sources. This is demonstrated in Fig. 4, where I plot the normalized magnitude of the inverse Fourier transform of the complex transfer function, which corresponds to the imaging of a very localized (ideally, delta-functional) near-field source. The half-power width of the focal spot (which corresponds to 0.707 level on the amplitude plot shown in Fig. 4) may be estimated in this case as about two array periods, for the both values of the loss tangent. Thus, the obtained lens resolution is about δ0 ≈ λh/5 ≈ λ0/7.3. The distance from the source plane to the image plane in this setup equals λh/2.
5. Conclusions In this work I have reviewed briefly the key principles of operation of the subwavelength imaging devices based on arrays of resonant inclusions. These principles have been supported by a number of theoretical and experimental works published during the last decade. Nevertheless, it appears that the potential of the very concept of the subwavelength imaging with just metasurfaces (both linear and nonlinear) as contrasted to volumetric metamaterials has yet to be fully realized, especially in the range of optical frequencies. Appendix A Electromagnetically, a dense array of dipolar particles (electric and magnetic) may be understood as a metasurface with the surface densities of the electric and magnetic dipole moments ps , ms induced by the electric and magnetic fields at the plane of the array: ps ¼ χ e
Et;1 þ Et;2 ; 2
ms ¼ χ m
Ht;1 þ Ht;2 ; 2
ðA:1Þ
where χ e ≡χ e ðω; kt Þ and χ m ≡χ m ðω; kt Þ are effective susceptibilities of the particles (that include the particle interaction in the array) to the fields in the middle plane of the array. The latter are the average of the tangential fields Et;1;2 and Ht;1;2 at the both sides of the array. The jumps of the tangential electric and magnetic fields across the metasurface are related to the surface densities of the electric and magnetic moments: Et;2 −Et;1 ¼ iωn ms ;
Ht;2 −Ht;1 ¼ −iωn ps ;
ðA:2Þ
where n is the unit normal to the metasurface. Splitting the total fields into the waves incident on the array and the waves scattered by the array and using the above relations, one obtains the transfer matrix of the array: Ta ¼
1 4−χ e χ m
ð2 þ χ e Þð2 þ χ m Þ −2ðχ e −χ m Þ
2ðχ e −χ m Þ ; ð2−χ e Þð2−χ m Þ
ðA:3Þ
−1
where χ e ¼ iωχ e ηp;s , χ m ¼ iωχ m ηp;s , and ηp,s are the characteristic impedances of the plane waves in the two orthogonal polarizations: pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi ηp ¼ μ 0 =ε0 kjj =k0 , ηs ¼ μ 0 =ε0 k0 =kjj . When χ e ¼ 2 and Fig. 4. The diffraction limited image of an ideal point source produced by a pair of plasmonic arrays with the complex transfer function T(ω0, ky) plotted in Fig. 3. This result is obtained by calculating the inverse Fourier transform of T(ω0, ky).
χ m ¼ −2, the array transfer matrix takes the form Ta = [0, 1; − 1, 0], and when χ e ¼ −2 and χ m ¼ 2 one obtains Ta = [0, − 1; 1, 0]. As one may see, a pair of such arrays satisfy the conditions of perfect lensing derived in Section 2.
S.I. Maslovski / Optics Communications 285 (2012) 3363–3367
To understand the physical meaning of these specific values of the susceptibilities, one may express χ e;m in terms of the standard polarizabilities αe,m ≡ αe,m(ω) of the particles to the local electric and magnetic fields: −1
−1
loc
ps ¼ A0 α e E ;
loc
ms ¼ A0 α m H ;
ðA:4Þ
where A0 is the unit cell area, and the local fields are loc
E
loc
H
¼
iωηp;s Et;1 þ Et;2 þ β e A0 − ps ; 2 2
! Ht;1 þ Ht;2 iω þ β m A0 − ¼ ms ; 2ηp;s 2
ðA:5Þ
where βe;m ≡βe;m ðω; kt Þ are the factors (interaction constants) that describe the interaction of the dipoles in the array. From here, after some algebra, χ e;m ¼
iω ; iω α −1 −β e;m þ 2 e;m
ðA:7Þ
−1 where α e ¼ α e A−1 α m ¼ α m A−1 β e ¼ βe A0 η−1 0 ηp;s , 0 ηp;s , p;s , and β m ¼ βm A0 ηp;s . Therefore, the situation when χ e;m ¼ 2 corresponds to −1
α e;m ðωÞ−β e;m ðω; kt Þ ¼ 0;
ðA:8Þ
which is the well-known dispersion equation for a surface mode (a surface plasmon-polariton) on a dense array of dipolar particles. Thus, in the two possible realizations discussed previously, either electric or magnetic subarray must be at resonance with an incoming wave: the transverse component of the wave vector of the incident wave must coincide with the propagation constant of a surface mode on the array. At the same time, another subarray with χ e;m ¼ −2 must be composed of the particles with the polarizability α e;m ðωÞ ¼
1 β e;m ðω; kt Þ−iω
:
resonators at microwave or even terahertz frequencies for this purpose (see, for instance, [24,25] where also the impact of particle interactions is studied in detail), at optical frequencies it is much harder to obtain a strong magnetic response. Therefore, it is desirable to get rid of the magnetic subsystem altogether. In this case, one may set χ m ¼ 0 and χ e ¼ 2 (that is, the same condition of the surface mode resonance is employed), for which the transfer matrix of the array equals Ta = [2, 1; − 1, 0]. When two such arrays are combined within a single device with the total transfer matrix given by Eq. (3), the scattering matrix of such a lens becomes
ðA:6Þ
ðA:9Þ
By tuning the polarizabilities of the particles, the Eqs. (A.8) and (A.9) can be always satisfied for a fixed pair of ðω; kt Þ. However, to be used in subwavelength imaging, the arrays must operate within a wide range of kt . Hence, it is beneficial to have the interaction constants of the array practically independent of kt . In arrays with central symmetry, β e;m ðω; −kt Þ ¼ β e;m ðω; kt Þ, and, therefore, at small transverse wavenumbers in an enough dense array one may have ∂β e;m =∂kt ≈0. In other words, the dispersion of the array surface modes must be very flat in kt, which is typical for plasmon-like waves. The disadvantage of the complete solution Eq. (A.3) is that it requires magnetically polarizable particles in addition to the electrically polarizable ones. While it is possible to employ split ring
3367
S0 ðω; kt Þ ¼
−t 21 =t 22 t 11 −t 12 t 21 =t 22
1=t 22 t 12 =t 22
¼−
1
−ikjj d
e
e−ikjj d ; ðA:10Þ 1
which shows that the evanescent waves transmitted through this device are amplified exactly as required for the perfect lens operation (notice the signs in the arguments of the exponents). However, in contrast to the ideal lens, this one also reflects the incoming evanescent waves with the reflection coefficient equal to − 1. Further details can be found in [11]. References [1] S. Lipson, H. Lipson, D. Tannhauser, Optical Physics, Cambridge, UK, 1998. [2] N. Engheta, R.W. Ziolkowski, Metamaterials: Physics and Engineering Explorations, John Wiley & Sons, 2006. [3] J.B. Pendry, Physical Review Letters 85 (2000) 3966. [4] V.G. Veselago, The electrodynamics of substances with simultaneously negative values of ε and μ 10 (1968) 509–514, (originally pubished in Russian in Uspekhi Fizicheskikh Nauk 92 (1967), 517–526). [5] R. Ruppin, Physics Letters A 277 (2000) 61. [6] X.S. Rao, C.K. Ong, Physical Review B 68 (2003) 113103. [7] S.I. Maslovski, S.A. Tretyakov, Journal of Applied Physics 94 (7) (2003) 4241. [8] J.B. Pendry, Science 322 (5898) (2008) 71. [9] A. Aubry, J.B. Pendry, Journal of the Optical Society of America B 27 (2010) 72. [10] P.-Y. Chen, A. Alù, Nano Letters 11 (12) (2011) 5514. [11] S.I. Maslovski, S.A. Tretyakov, P. Alitalo, Journal of Applied Physics 96 (3) (2004) 1293. [12] C. Simovski, A. Viitanen, S. Tretyakov, Physical Review E 72 (2005) 066606. [13] P. Alitalo, S. Maslovski, S. Tretyakov, Physics Letters A 357 (4–5) (2006) 397. [14] P. Alitalo, C. Simovski, A. Viitanen, S. Tretyakov, Physical Review B 74 (2006) 235425. [15] C. Simovski, A. Viitanen, S. Tretyakov, Journal of Applied Physics 101 (2007) 123102. [16] C. Simovski, S. Tretyakov, A. Viitanen, Technical Physics Letters 33 (2007) 264. [17] S. Maslovski, P. Alitalo, S. Tretyakov, Journal of Applied Physics 104 (10) (2008) 103109. [18] C. Mateo-Segura, C.R. Simovski, G. Goussetis, S. Tretyakov, Optics Letters 34 (2009) 2333. [19] A.L. Fructos, S. Campione, F. Capolino, F. Mesa, Journal of the Optical Society of America B 28 (2011) 1446. [20] M.J. Freire, R. Marqués, Applied Physics Letters 86 (18) (2005) 182505. [21] X. Huang, S. Xiao, D. Ye, J. Huangfu, Z. Wang, L. Ran, L. Zhou, Optics Express 18 (10) (2010) 10377. [22] M.A. Ordal, R.J. Bell, J.R.W. Alexander, L.L. Long, M.R. Querry, Applied Optics 24 (24) (1985) 4493. [23] P.B. Johnson, R.W. Christy, Physical Review B 6 (12) (1972) 4370. [24] R. Singh, C. Rockstuhl, F. Lederer, W. Zhang, Applied Physics Letters 94 (2009) 021116. [25] R. Singh, C. Rockstuhl, W. Zhang, Applied Physics Letters 97 (2010) 241108.
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Optics Communications journal homepage: www.elsevier.com/locate/optcom
Subwavelength imaging with arrays of plasmonic scatterers Stanislav I. Maslovski Departamento de Engenharia Electrotécnica, Instituto de Telecomunicações, Universidade de Coimbra, Pólo II, 3030-290 Coimbra, Portugal
a r t i c l e
i n f o
Article history: Received 7 November 2011 Received in revised form 20 December 2011 Accepted 21 December 2011 Available online 3 January 2012
a b s t r a c t Here, I briefly review the key principles of subwavelength imaging in dense arrays of resonant scatterers, with an emphasis on resonant plasmonic structures at optical frequencies. © 2012 Elsevier B.V. All rights reserved.
Keywords: Subwavelength imaging Plasmonic resonance Dipolar arrays
1. Introduction The resolving power of conventional optical instruments such as microscopes is limited by the Abbe diffraction limit [1]. The nature of this limit becomes evident in the Fourier optics when the optical field is decomposed into components with different spatial frequencies, that is, with different values of the transverse wavenumber kt. Conventional lenses act as filters on this spatial spectrum, which select only the components within a limited range of wavenumbers: 0 ≤ kt ≤ kt, max b 2π/λ. Therefore, the spatial resolution of such devices is limited by δ0 ≈ π/kt, max. With the advent of metamaterials [2] it became possible to widen the spectrum of spatial frequencies to which an optical instrument may react and in this way to increase the resolving power. Namely, if optical resolution better than a half wavelength is required, it is necessary for an optical device to react to the components of the source field characterized by the wavenumbers kt > 2π/λ. These waves exist only in the near field of a source, as they decay exponentially away from the source. While such evanescent components may be picked up with a closely located sensor of the characteristic dimension on the order of the required resolution, it may be as well done with metamaterial slabs or metasurfaces that support oscillatory states localized on the scale of the required resolution. The well-known theoretical concept of perfect lensing [3] with a planar slab of a double negative metamaterial characterized by εr = − 1 and μr = − 1 (at a certain frequency of operation) belongs to the latter category, because the free surface of such ideal Veselago medium [4] supports surface plasmon-polaritons with arbitrary wavenumbers kt [5,6], and, thus, localized oscillations of arbitrarily small spatial scale can be excited on this surface. As discussed ahead, a similar behavior can be realized without employing any volumetric metamaterials. For example, one may obtain an analogous effect with sheets of nonlinear media that realize the phase 0030-4018/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2011.12.079
conjugation operation [7–10] or, under certain limitations, with arrays of weakly interacting and linear resonant inclusions (which may be plasmonic particles in optics or metallic inclusions at microwaves) [11–19]. In what follows, I review the principles of operation of such linear metasurfaces capable of picking up and restoring the evanescent components of the source field. 2. Subwavelength imaging in arrays of resonant inclusions It is convenient to start with identifying the general requirements on the transfer function of an optical system that is capable of resolving details much finer than a half wavelength. The transfer function T(ω, kt) can be formally introduced as a coefficient of proportionality between the field components of interest at the source plane and at the image plane of an imaging system. With this definition at hand, the time-harmonic optical electric field (here I consider a fixed polarization and work with scalar field) in the image plane can be written as 2
Eðω; rt Þ ¼ ∫∫
d kt ik ·r 2 ′ src −ik ·r′ T ðω; kt Þe t t ∫∫d r t E ðω; rt Þet t ; ð2πÞ2
ð1Þ
where Esrc ðω; rt Þ is the field at the source plane, with rt and r′t being the radius vectors in the image and the source planes, respectively. I use the time dependence of the form exp(− iωt). Evidently, Eðω; rt Þ ¼ Esrc ðω; rt Þ when T ðω; kt Þ≡1. This is the case of the perfect lens that restores all spectral components of the source field at the image plane, among which there are the propagating waves with jkt j b 2π=λ, and the evanescent waves with jkt j > 2π=λ. Typically, the source is placed at a certain distance from the frontal surface of the lens. There is as well a finite separation from the back side of the lens to the focal plane. Therefore, the total transfer
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Fig. 1. Left: A pair of two-dimensional arrays of resonant particles (plasmonic spheres) separated by distance d, operating as a subwavelength imaging device. Right: Typical dispersion curve for a proper surface mode on an array of plasmonic spheres which is required for subwavelength imaging at the frequency ω0.
function T ðω; kt Þ is, generally, a product of three components that take into account the propagation in the two gaps before and after the lens, and the propagation in the lens itself. In order for the above interpretation to be rigorous, the scalar transfer functions must be replaced with transfer matrices of the air gaps and the lens [11]. Then multiplying these matrices sequentially produces the total transfer matrix. Assuming the same thickness d/2 of the two gaps, one may write for the total transfer matrix ikjj d
T ðω; kt Þ ¼
e
2
! 0ik
ikjj d
·T 0 ðω; kt Þ· e
2
! 0ik
; ð2Þ jj d jj d 0 e− 2 0 e− 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where kjj ¼ ðω=cÞ2 −k2t is the propagation factor along the optical axis of the imaging system. The matrix T 0 ðω; kt Þ represents the transfer matrix of the lens. Evidently, for the complete device to operate as a perfect lens this matrix must be the inverse of the product of the two matrices of the air gaps. In this case the total transfer matrix of the system becomes an identity matrix. As is known, a planar slab of ideal Veselago medium [3,4] with εr = μr = − 1 and the thickness d has the required property, as it essentially negates the effect of wave propagation in a free space layer of the same thickness, and, thus, can be used as the middle component in Eq. (2). However, there is a possibility to obtain a similar effect without employing any volumetric metamaterials. One may notice that the important physical phenomena, namely, the negative refraction of the incident propagating waves and the resonant excitation of coupled surface plasmon-polaritons under the incidence of the evanescent waves happen at the two surfaces of the Veselago slab. Therefore, one might wish to examine a system composed of two metasurfaces separated with just air and look for the characteristics of these surfaces such that the total system would resemble a perfect lens. In this way we found in [7] that a pair of surfaces that phase-conjugate both electric and magnetic fields in air has the required properties of a perfect lens. Later other authors proposed devices based on similar principles [8–10]. The phase conjugation (time reversal) operation requires nonlinearities and wave mixing which I do not consider here. However, if one is interested only in the evanescent part of the spectrum, the necessary properties may be obtained without employing any nonlinearities. One may start with the transfer matrix of the lens in the form [11] T 0 ðω; kt Þ ¼
a~ c~
b~ ⋅ eikjj d ~ d 0
0
−ikjj d
e
e~ ⋅ g~
f~ ; ~ h
ð3Þ
where the two matrices on the left and on the right with yet unknown components represent the metasurfaces. What must be the components of these matrices so that the transfer matrix T 0 ðω; kt Þ becomes the inverse of the transfer matrix of an air layer, and, thus, the imaging system acquires the perfect lens
property? One may verify that this happens when a~ ¼ d~ ¼ 0, e~ ¼ h~ ¼ 0, and b~ g~ ¼ c~f~ ¼ 1. Quite interestingly, such an abstract mathematical result that follows just from the rules of matrix multiplication is actually physically realizable with dense two-dimensional arrays of resonant inclusions when they are excited by evanescent waves (see Fig. 1). By analogy with the operation of Pendry's lens [3], subwavelength imaging in such arrays occurs when they support surface modes (surface plasmon-polaritons) with the propagation factors matching the transverse wavenumbers of the incident evanescent waves. As explained in Appendix A, when a pair of arrays separated by distance d is under incidence of an evanescent wave with the frequency ω and the transverse wave vector kt that satisfy the dispersion relation for the surface modes on the arrays α
−1
ðωÞ−βðω; kt Þ ¼ 0;
ð4Þ
where α(ω) is the inclusion polarizability and βðω; kt Þ is the interaction constant of the array, the transmission coefficient through the array pair equals s21 ðω; kt Þ ¼ −exp −ikjj d for this wave. Thus, the evanescent components are amplified when tunneled through the arrays, so that the decay of the same waves in the regions before and after the arrays gets compensated. This effect was theoretically explained and experimentally verified for arrays of electrically polarizable inclusions at microwaves in our initial works [11,13] and, for magnetically polarizable inclusions, in [20]. One may notice that at a given frequency ω the dispersion relation Eq. (4) is satisfied only at a certain kt. Nevertheless, when the dispersion of the surface waves is enough flat in kt (such as shown in Fig. 1, right), practical examples show that the amplification of the evanescent waves may happen in a wide range of wavenumbers that is enough for subwavelength imaging at a fixed frequency. 3. Imaging in arrays of plasmonic particles and related challenges The realizations based on arrays with both electric and magnetic response considered in [11] (see also Appendix A) are idealized structures that aim at both the subwavelength resolution and the matching (that is, the absence of reflection). When the level of reflection is not important, one may employ the particles with only electric response. At optical and near-infrared frequencies a natural choice for such inclusions may be nanoparticles made of noble metals. In the optical band such metals exhibit plasmonic behavior due to the material response which is typically described by the Drude dispersion [22]: εr ¼ ε∞ −
ω2p ; ωðω þ iΓ Þ
ð5Þ
where ωp is the plasma frequency of the electrons in the metal, and Γ is the collision frequency that represents the loss. More precise models include additional Lorentz-type dispersion terms.
S.I. Maslovski / Optics Communications 285 (2012) 3363–3367
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The resonant frequency of a small spherical nanoparticle embedded into a dielectric with the relative permittivity εh may be estimated from the relation for the quasi-static polarizability of such a particle: 3
α ¼ 4πεh R
εr −εh ; εr þ 2ε h
ð6Þ
where R is the radius of the nanoparticle. Neglecting the effect of loss, the polarizability resonates at the frequency ω0 where εr(ω0) ≈ − 2εh. For more precise formulation one may use the well-known Mie theory. Over the last decade, subwavelength imaging with arrays of such nanoparticles was successfully demonstrated in a number of works [12,14–19]. In order to identify which structures are more promising for the subwavelength imaging purposes, one must study the surface waves supported by the structures. For instance, in [19] the dispersion of surface modes of several types and polarizations on a rectangular array of plasmonic spheres was considered. Some search for structures with optimal particle polarizabilities and flat dispersion of the surface waves was performed in [15,18]. On the other hand, in [17] the frequency scanning-based subwavelength imaging was proposed, for the plasmonic arrays with profound dispersion. In most of the mentioned works the nanoparticles are typically made of silver, that has the loss tangent |εr″/ε′r| ≈ 10 − 2 … 10 − 1 in the optical and near infrared bands [22]. For the distance between the arrays on the order of a half wavelength, this amount of loss limits the resolution by δ0 ≈ λ/5 … λ/4 (at terahertz or microwave frequencies the loss in metal particles is much lower and the resolution is limited by the period of the arrays mostly). The resolution can be improved in more closely positioned arrays, or in arrays embedded into an active host that compensates the loss. Aperiodicity in the positioning of the particles in the arrays may affect the resolution in the same manner as loss, due to random scattering. This effect has not yet been thoroughly studied. The finite size of the arrays has an insignificant influence on the subwavelength imaging performance when compared to the effect of loss. For instance, in [14] the arrays of 20 × 20 plasmonic spheres have been studied. While a considerable difference in the optimal operational frequency of the lens was observed (which is related to a different effective value of the interaction constant βðω; kt Þ in a finite array when compared to an infinite array), the obtained results generally follow the predictions of the theory initially developed for the infinite arrays. The examples available from the literature show that the surface modes polarized orthogonally to the plane of the array have a dispersion with less pronounced dependence on kt than the modes with the nanoparticles polarized tangentially to the array plane, and, thus, may resonate with the incident evanescent waves in a broader range of wavenumbers. Nevertheless, in a relatively recent paper [18], using the particles of oblate spheroidal shape a resolution on the order of λ/5 (with the separation between the image and the source plane not less than a half wavelength) was demonstrated for both main polarizations. Although in this paper I concentrate on imaging with arrays of plasmonic scatterers, the same principles apply to complementary structures where the particles are replaced with holes in a sheet of plasmonic metal. For example, in [21], Huang et al. proposed a nearfield lens formed by an array of fractal-shaped holes in a metal plate. Completely analogous to the cases considered above, the authors of [21] aimed for a flat dispersion of the surface modes (surface plasmon-polaritons) of both main polarizations, which was optimal for the subwavelength imaging. They reported such flat bands of a structured silver film at the wavelengths of about 3750 nm (80 THz) and 7500 nm (40 THz), at which |ε′r| of silver is on the order of 10 3, that is, it already approaches the properties of a perfect electric conductor. Therefore, the resolution in this type of lens is limited mostly by the structural period (which is on the order of λ/7 in their case), as in the early microwave experiments [11,13,20].
Fig. 2. The unit cell of a double array of plasmonic particles, as simulated in CST Microwave Studio™. The structure is periodic along the x and y axes with the period a.
However, at optical wavelengths the situation is significantly different: typically, one has to reduce the amount of metal in a structure to obtain a higher quality factor and better resolution. In the next section I consider such a situation where the silver nanoparticles are “shrunk” in the directions parallel to the array plane, which in the end may allow for a resolution on the order of the same λ0/7, but in the visible. 4. Numerical example: imaging in arrays of prolate spheroidal plasmonic nanoparticles Here, I consider subwavelength imaging in coupled arrays of plasmonic prolate spheroids. The geometry of the spheroids is shown in Fig. 2. The spheroids are elongated along the axis orthogonal to the array plane (the z-axis). As discussed in the previous section, such orientation allows for a flat dispersion with respect to the transverse component of the wave vector. The prolate spheroidal plasmonic inclusions may also have a significantly lower resonant frequency. Namely, in the quasi-static approximation the polarizability of such particles along the longest axis reads ε h ðεr −εh Þ ; εh þ Nðεr −εh Þ
α¼V
ð7Þ
where V is the inclusion volume and N is the depolarization factor N¼
1 −1 e2
1 1þe ln −1 ; 2e 1−e
ð8Þ
pffiffiffiffiffiffiffiffiffiffiffiffi where e ¼ 1−r 2 is the eccentricity that is calculated from the equatorial-to-polar axis ratio r of the ellipsoid. For prolate spheroids with r = 1/3, N ≈ 1/10, and, therefore, the plasmonic resonance of an isolated nanoparticle occurs at the frequency where εr ðω0 Þ≈− 1−N N ε h ≈−9ε h . From here, for silver nanoparticles with ωp ≈ 2π × 2.18 × 103 THz (Ref. [22]) embedded into a host with εh = 2.12 (e.g., silicon dioxide), one obtains from Eq. (5) that the plasmonic resonance red-shifts in this case from the near ultraviolet to the visible and occurs at the vacuum wavelength of about 6.5 × 10 2 nm. When the nanoparticles are assembled into arrays, the resonance may experience a further shift, because of the particle interactions. Namely, full-wave simulations of the structure depicted in Fig. 2 demonstrate that in arrays of prolate spheroidal silver nanoparticles with r = 1/3, R = 2a/3 and the array period a = λh/10 with λh ¼ pffiffiffiffiffi λ0 = εh and εh = 2.12, the plasmonic resonance occurs at around λ0 = 660 nm. At this wavelength (hν = 1.88 eV) the complex permittivity of silver is εr′ ¼ −20 0:1 and εr″ ¼ 0:45 0:18 [23], thus, the loss tangent of the material of the nanoparticles is within 0.013 b tan
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Fig. 3. The complex transfer function T(ω0, ky) of a pair of arrays of plasmonic silver particles with the period a = 45.3 nm separated by the distance d = 2.5a. The absolute value of T (on the left) and the argument of T (on the right) are plotted as functions of the normalized transverse wave vector component kya, for the indicated values of the loss tangent of the plasmonic metal. The array particles are prolate spheroids with the polar radius R = 2a/3 and the equatorial radius of R/3. The particles are embedded in a dielectric with permittivity εh = 2.12. The wavelength of operation (in vacuum) is λ0 = 660 nm.
δ b 0.032. The loss in the host dielectric is typically much lower and may be neglected. In order to demonstrate the influence of loss on the subwavelength imaging in arrays of silver nanoparticles, I solve with full-wave simulations for the total complex transfer function (including the air gaps) T(ω0, ky) of a pair of such arrays separated by d = λh/4. These results are shown in Fig. 3 for the case of P-polarized incident light with the transverse wave vector kt oriented along the y-axis. Assuming that the loss is at the lower limit, I obtain a transfer function that over amplifies the evanescent components with kya ≈ 1.2 (the dashed blue curve). On the contrary, when the loss is at the upper limit, one may observe a significant dumping of the components with kya > 1.4 as compared to the case of low loss (the solid green curve). One may also notice a dip on these plots at kya = 2πa/λh, which is at the boundary between the propagating and the evanescent part of the spatial spectrum. In both considered cases, the transmitted spectrum is practically completely cut off at around kya = 2. Nevertheless, besides the mentioned non-idealities in the transfer function, such a near-field lens is actually able to produce rather sharp images of point sources. This is demonstrated in Fig. 4, where I plot the normalized magnitude of the inverse Fourier transform of the complex transfer function, which corresponds to the imaging of a very localized (ideally, delta-functional) near-field source. The half-power width of the focal spot (which corresponds to 0.707 level on the amplitude plot shown in Fig. 4) may be estimated in this case as about two array periods, for the both values of the loss tangent. Thus, the obtained lens resolution is about δ0 ≈ λh/5 ≈ λ0/7.3. The distance from the source plane to the image plane in this setup equals λh/2.
5. Conclusions In this work I have reviewed briefly the key principles of operation of the subwavelength imaging devices based on arrays of resonant inclusions. These principles have been supported by a number of theoretical and experimental works published during the last decade. Nevertheless, it appears that the potential of the very concept of the subwavelength imaging with just metasurfaces (both linear and nonlinear) as contrasted to volumetric metamaterials has yet to be fully realized, especially in the range of optical frequencies. Appendix A Electromagnetically, a dense array of dipolar particles (electric and magnetic) may be understood as a metasurface with the surface densities of the electric and magnetic dipole moments ps , ms induced by the electric and magnetic fields at the plane of the array: ps ¼ χ e
Et;1 þ Et;2 ; 2
ms ¼ χ m
Ht;1 þ Ht;2 ; 2
ðA:1Þ
where χ e ≡χ e ðω; kt Þ and χ m ≡χ m ðω; kt Þ are effective susceptibilities of the particles (that include the particle interaction in the array) to the fields in the middle plane of the array. The latter are the average of the tangential fields Et;1;2 and Ht;1;2 at the both sides of the array. The jumps of the tangential electric and magnetic fields across the metasurface are related to the surface densities of the electric and magnetic moments: Et;2 −Et;1 ¼ iωn ms ;
Ht;2 −Ht;1 ¼ −iωn ps ;
ðA:2Þ
where n is the unit normal to the metasurface. Splitting the total fields into the waves incident on the array and the waves scattered by the array and using the above relations, one obtains the transfer matrix of the array: Ta ¼
1 4−χ e χ m
ð2 þ χ e Þð2 þ χ m Þ −2ðχ e −χ m Þ
2ðχ e −χ m Þ ; ð2−χ e Þð2−χ m Þ
ðA:3Þ
−1
where χ e ¼ iωχ e ηp;s , χ m ¼ iωχ m ηp;s , and ηp,s are the characteristic impedances of the plane waves in the two orthogonal polarizations: pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi ηp ¼ μ 0 =ε0 kjj =k0 , ηs ¼ μ 0 =ε0 k0 =kjj . When χ e ¼ 2 and Fig. 4. The diffraction limited image of an ideal point source produced by a pair of plasmonic arrays with the complex transfer function T(ω0, ky) plotted in Fig. 3. This result is obtained by calculating the inverse Fourier transform of T(ω0, ky).
χ m ¼ −2, the array transfer matrix takes the form Ta = [0, 1; − 1, 0], and when χ e ¼ −2 and χ m ¼ 2 one obtains Ta = [0, − 1; 1, 0]. As one may see, a pair of such arrays satisfy the conditions of perfect lensing derived in Section 2.
S.I. Maslovski / Optics Communications 285 (2012) 3363–3367
To understand the physical meaning of these specific values of the susceptibilities, one may express χ e;m in terms of the standard polarizabilities αe,m ≡ αe,m(ω) of the particles to the local electric and magnetic fields: −1
−1
loc
ps ¼ A0 α e E ;
loc
ms ¼ A0 α m H ;
ðA:4Þ
where A0 is the unit cell area, and the local fields are loc
E
loc
H
¼
iωηp;s Et;1 þ Et;2 þ β e A0 − ps ; 2 2
! Ht;1 þ Ht;2 iω þ β m A0 − ¼ ms ; 2ηp;s 2
ðA:5Þ
where βe;m ≡βe;m ðω; kt Þ are the factors (interaction constants) that describe the interaction of the dipoles in the array. From here, after some algebra, χ e;m ¼
iω ; iω α −1 −β e;m þ 2 e;m
ðA:7Þ
−1 where α e ¼ α e A−1 α m ¼ α m A−1 β e ¼ βe A0 η−1 0 ηp;s , 0 ηp;s , p;s , and β m ¼ βm A0 ηp;s . Therefore, the situation when χ e;m ¼ 2 corresponds to −1
α e;m ðωÞ−β e;m ðω; kt Þ ¼ 0;
ðA:8Þ
which is the well-known dispersion equation for a surface mode (a surface plasmon-polariton) on a dense array of dipolar particles. Thus, in the two possible realizations discussed previously, either electric or magnetic subarray must be at resonance with an incoming wave: the transverse component of the wave vector of the incident wave must coincide with the propagation constant of a surface mode on the array. At the same time, another subarray with χ e;m ¼ −2 must be composed of the particles with the polarizability α e;m ðωÞ ¼
1 β e;m ðω; kt Þ−iω
:
resonators at microwave or even terahertz frequencies for this purpose (see, for instance, [24,25] where also the impact of particle interactions is studied in detail), at optical frequencies it is much harder to obtain a strong magnetic response. Therefore, it is desirable to get rid of the magnetic subsystem altogether. In this case, one may set χ m ¼ 0 and χ e ¼ 2 (that is, the same condition of the surface mode resonance is employed), for which the transfer matrix of the array equals Ta = [2, 1; − 1, 0]. When two such arrays are combined within a single device with the total transfer matrix given by Eq. (3), the scattering matrix of such a lens becomes
ðA:6Þ
ðA:9Þ
By tuning the polarizabilities of the particles, the Eqs. (A.8) and (A.9) can be always satisfied for a fixed pair of ðω; kt Þ. However, to be used in subwavelength imaging, the arrays must operate within a wide range of kt . Hence, it is beneficial to have the interaction constants of the array practically independent of kt . In arrays with central symmetry, β e;m ðω; −kt Þ ¼ β e;m ðω; kt Þ, and, therefore, at small transverse wavenumbers in an enough dense array one may have ∂β e;m =∂kt ≈0. In other words, the dispersion of the array surface modes must be very flat in kt, which is typical for plasmon-like waves. The disadvantage of the complete solution Eq. (A.3) is that it requires magnetically polarizable particles in addition to the electrically polarizable ones. While it is possible to employ split ring
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S0 ðω; kt Þ ¼
−t 21 =t 22 t 11 −t 12 t 21 =t 22
1=t 22 t 12 =t 22
¼−
1
−ikjj d
e
e−ikjj d ; ðA:10Þ 1
which shows that the evanescent waves transmitted through this device are amplified exactly as required for the perfect lens operation (notice the signs in the arguments of the exponents). However, in contrast to the ideal lens, this one also reflects the incoming evanescent waves with the reflection coefficient equal to − 1. Further details can be found in [11]. References [1] S. Lipson, H. Lipson, D. Tannhauser, Optical Physics, Cambridge, UK, 1998. [2] N. Engheta, R.W. Ziolkowski, Metamaterials: Physics and Engineering Explorations, John Wiley & Sons, 2006. [3] J.B. Pendry, Physical Review Letters 85 (2000) 3966. [4] V.G. Veselago, The electrodynamics of substances with simultaneously negative values of ε and μ 10 (1968) 509–514, (originally pubished in Russian in Uspekhi Fizicheskikh Nauk 92 (1967), 517–526). [5] R. Ruppin, Physics Letters A 277 (2000) 61. [6] X.S. Rao, C.K. Ong, Physical Review B 68 (2003) 113103. [7] S.I. Maslovski, S.A. Tretyakov, Journal of Applied Physics 94 (7) (2003) 4241. [8] J.B. Pendry, Science 322 (5898) (2008) 71. [9] A. Aubry, J.B. Pendry, Journal of the Optical Society of America B 27 (2010) 72. [10] P.-Y. Chen, A. Alù, Nano Letters 11 (12) (2011) 5514. [11] S.I. Maslovski, S.A. Tretyakov, P. Alitalo, Journal of Applied Physics 96 (3) (2004) 1293. [12] C. Simovski, A. Viitanen, S. Tretyakov, Physical Review E 72 (2005) 066606. [13] P. Alitalo, S. Maslovski, S. Tretyakov, Physics Letters A 357 (4–5) (2006) 397. [14] P. Alitalo, C. Simovski, A. Viitanen, S. Tretyakov, Physical Review B 74 (2006) 235425. [15] C. Simovski, A. Viitanen, S. Tretyakov, Journal of Applied Physics 101 (2007) 123102. [16] C. Simovski, S. Tretyakov, A. Viitanen, Technical Physics Letters 33 (2007) 264. [17] S. Maslovski, P. Alitalo, S. Tretyakov, Journal of Applied Physics 104 (10) (2008) 103109. [18] C. Mateo-Segura, C.R. Simovski, G. Goussetis, S. Tretyakov, Optics Letters 34 (2009) 2333. [19] A.L. Fructos, S. Campione, F. Capolino, F. Mesa, Journal of the Optical Society of America B 28 (2011) 1446. [20] M.J. Freire, R. Marqués, Applied Physics Letters 86 (18) (2005) 182505. [21] X. Huang, S. Xiao, D. Ye, J. Huangfu, Z. Wang, L. Ran, L. Zhou, Optics Express 18 (10) (2010) 10377. [22] M.A. Ordal, R.J. Bell, J.R.W. Alexander, L.L. Long, M.R. Querry, Applied Optics 24 (24) (1985) 4493. [23] P.B. Johnson, R.W. Christy, Physical Review B 6 (12) (1972) 4370. [24] R. Singh, C. Rockstuhl, F. Lederer, W. Zhang, Applied Physics Letters 94 (2009) 021116. [25] R. Singh, C. Rockstuhl, W. Zhang, Applied Physics Letters 97 (2010) 241108.