OSA's Digital Library

Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 12 — Jun. 17, 2013
  • pp: 15026–15036
« Show journal navigation

Hyperbolic metamaterial lens with hydrodynamic nonlocal response

Wei Yan, N. Asger Mortensen, and Martijn Wubs  »View Author Affiliations


Optics Express, Vol. 21, Issue 12, pp. 15026-15036 (2013)
http://dx.doi.org/10.1364/OE.21.015026


View Full Text Article

Acrobat PDF (1369 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

Abstract

We investigate the effects of hydrodynamic nonlocal response in hyperbolic metamaterials (HMMs), focusing on the experimentally realizable parameter regime where unit cells are much smaller than an optical wavelength but much larger than the wavelengths of the longitudinal pressure waves of the free-electron plasma in the metal constituents. We derive the nonlocal corrections to the effective material parameters analytically, and illustrate the noticeable nonlocal effects on the dispersion curves numerically. As an application, we find that the focusing characteristics of a HMM lens in the local-response approximation and in the hydrodynamic Drude model can differ considerably. In particular, the optimal frequency for imaging in the nonlocal theory is blueshifted with respect to that in the local theory. Thus, to detect whether nonlocal response is at work in a hyperbolic metamaterial, we propose to measure the near-field distribution of a hyperbolic metamaterial lens.

© 2013 osa

1. Introduction

Hyperbolic metamaterials (HMMs), also known as indefinite media, enjoy a great deal of attention owing to their unique hyperbolic dispersion relations [1

1. D. R. Smith and D. Schurig, “Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors,” Phys. Rev. Lett. 90, 077405 (2003) [CrossRef] [PubMed] .

14

14. M. Yan and N. A. Mortensen, “Hollow-core infrared fiber incorporating metal-wire metamaterial,” Opt. Express 17, 14851–14864 (2009) [CrossRef] [PubMed] .

], with associated high-wavenumber propagating waves without upper limit. This leads to numerous applications, such as enhanced-light interactions [4

4. Z. Jacob, I. Smolyaninov, and E. Narimanov, “Broadband Purcell effect: Radiative decay engineering with metamaterials,” Appl. Phys. Lett. 100, 181105 (2012).

7

7. M. A. Noginov, Y. A. Barnakov, G. Zhu, T. Tumkur, H. Li, and E. E. Narimanov, “Bulk photonic metamaterial with hyperbolic dispersion,” Appl. Phys. Lett. 94, 151105 (2009) [CrossRef] .

], subwavelength imaging [8

8. P. Belov and Y. Hao, “Subwavelength imaging at optical frequencies using a transmission device formed by a periodic layered metal-dielectric structure operating in the canalization regime,” Phys. Rev. B 73, 113110 (2006) [CrossRef] .

13

13. Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315, 1686 (2007) [CrossRef] [PubMed] .

], negative refraction [1

1. D. R. Smith and D. Schurig, “Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors,” Phys. Rev. Lett. 90, 077405 (2003) [CrossRef] [PubMed] .

], and low-loss fiber cladding [14

14. M. Yan and N. A. Mortensen, “Hollow-core infrared fiber incorporating metal-wire metamaterial,” Opt. Express 17, 14851–14864 (2009) [CrossRef] [PubMed] .

]. HMMs are now also be emphasized for their potential applications in quantum nanophotonics [15

15. C. L. Cortes, W. Newman, S. Molesky, and Z. Jacob, “Quantum nanophotonics using hyperbolic metamaterials,” J. Opt. 14, 063001 (2012) [CrossRef] .

]. The HMMs are usually artificially made by periodic dielectric-metal structures, such as a 1D dielectric-metal Bragg grating. To describe the optical properties of the HMMs, it is common to employ the local-response approximation (LRA), i.e. with every location r in the structure a certain value for the permittivity ε(r) is associated. In the LRA, the effective material parameters of HMMs have been thoroughly studied [16

16. J. Elser, V. Podolksiy, I. Salakhutdinov, and I. Avrutsky, “Nonlocal effects in effective-medium response of nanolayered metamaterials,” Appl. Phys. Lett. 90, 191109 (2007).

20

20. L. Shen, T. Yang, and Y. Chau, “50/50 beam splitter using a one-dimensional metal photonic crystal with parabo-lalike dispersion,” Appl. Phys. Lett. 90, 251909 (2007) [CrossRef] .

].

Thanks to advances in nanofabrication, we witness a miniaturization of the feature size of metamaterials towards the deep nanoscale. The LRA becomes more inaccurate, since the nonlocal response of free electrons starts to play a role [21

21. F. Bloch, “Bremsvermögen von Atomen mit mehreren Elektronen,” Z. Phys. A 81, 363–376 (1933).

33

33. W. Yan, M. Wubs, and N. A. Mortensen, “Hyperbolic metamaterials: nonlocal response regularizes broadband supersingularity,” Phys. Rev. B 86, 205429 (2012) [CrossRef] .

]. It is known that the nonlocal response causes a blueshift of the surface plasmon (SP) resonance of the metallic particle [26

26. S. Raza, G. Toscano, A.-P. Jauho, M. Wubs, and N. A. Mortensen, “Unusual resonances in nanoplasmonic structures due to nonlocal response,” Phys. Rev. B 84, 121412(R) (2011) [CrossRef] .

29

29. S. Raza, N. Stenger, S. Kadkhodazadeh, S. V. Fischer, N. Kostesha, A.-P. Jauho, A. Burrows, M. Wubs, and N. A. Mortensen, “Blueshift of the surface plasmon resonance in silver nanoparticles studied with EELS,” Nanophotonics 2, 131 (2013) [CrossRef] .

], and limits the SP field enhancement that in the LRA sometimes diverges [30

30. G. Toscano, S. Raza, A.-P. Jauho, M. Wubs, and N. A. Mortensen, “Modified field enhancement and extinction in plasmonic nanowire dimers due to nonlocal response,” Opt. Express 13, 4176–4188 (2012) [CrossRef] .

32

32. A. I. Fernández-Domínguez, A. Wiener, F. J. García-Vidal, S. A. Maier, and J. B. Pendry, “Transformation-optics description of nonlocal effects in plasmonic nanostructures,” Phys. Rev. Lett. 108, 106802 (2012) [CrossRef] [PubMed] .

]. We employ a simple generalization to the LRA, namely the hydrodynamic Drude model (HDM) [21

21. F. Bloch, “Bremsvermögen von Atomen mit mehreren Elektronen,” Z. Phys. A 81, 363–376 (1933).

, 22

22. A. D. Boardman, Electromagnetic Surface Modes (John Wiley and Sons, Chichester, 1982).

], which takes the nonlocal response into account. New in this model, as compared to the LRA, are longitudinal waves with sub-nanometer wavelengths, besides the usual transverse waves.

Incidentally, some HMMs have been found to exhibit strong effective ‘nonlocal response’, even in studies that employ the local-response approximation [16

16. J. Elser, V. Podolksiy, I. Salakhutdinov, and I. Avrutsky, “Nonlocal effects in effective-medium response of nanolayered metamaterials,” Appl. Phys. Lett. 90, 191109 (2007).

20

20. L. Shen, T. Yang, and Y. Chau, “50/50 beam splitter using a one-dimensional metal photonic crystal with parabo-lalike dispersion,” Appl. Phys. Lett. 90, 251909 (2007) [CrossRef] .

]. What is meant here is that the HMMs, when considered as scatterers, cannot be described in the single-scattering Born approximation. This differs from the (material) nonlocal response of HMMs that we consider here and in Ref. [33

33. W. Yan, M. Wubs, and N. A. Mortensen, “Hyperbolic metamaterials: nonlocal response regularizes broadband supersingularity,” Phys. Rev. B 86, 205429 (2012) [CrossRef] .

], where already the scattering potentials associated with the metal constituents are nonlocal.

2. Dispersion relations of hyperbolic metamaterials

As the HMM we consider a 1D subwavelength dielectric-metal Bragg grating, with a unit cell of thickness d, and thicknesses a and b of the dielectric and metal layers, respectively. The permittivity of the dielectric layer is εd. The metal is described in the HDM as a free-electron plasma with [21

21. F. Bloch, “Bremsvermögen von Atomen mit mehreren Elektronen,” Z. Phys. A 81, 363–376 (1933).

, 22

22. A. D. Boardman, Electromagnetic Surface Modes (John Wiley and Sons, Chichester, 1982).

, 26

26. S. Raza, G. Toscano, A.-P. Jauho, M. Wubs, and N. A. Mortensen, “Unusual resonances in nanoplasmonic structures due to nonlocal response,” Phys. Rev. B 84, 121412(R) (2011) [CrossRef] .

]
εmT(ω)=1ωp2ω2+iωγ,εmL(k,ω)=1ωp2ω2+iωγβ2k2,
(1)
where εmT is the Drude permittivity for the transverse electric fields as in the LRA, while the wavevector dependence of the permittivity εmL of the longitudinal electric fields is responsible for the nonlocal response.

We apply the hydrodynamic generalization of the common transfer-matrix method for layered systems [23

23. W. L. Mochán, M. Castillo-Mussot, and R. G. Barrera, “Effect of plasma waves on the optical properties of metal-insulator superlattices,” Phys. Rev. B 15, 1088–1098 (1987) [CrossRef] .

, 33

33. W. Yan, M. Wubs, and N. A. Mortensen, “Hyperbolic metamaterials: nonlocal response regularizes broadband supersingularity,” Phys. Rev. B 86, 205429 (2012) [CrossRef] .

], and obtain the exact dispersion equation of the HMM
cosθb={cosθd[kmLcosθmsinθlk||(wdwm)zmsinθmcosθl]+k||(wdwm)zdsinθd(1cosθmcosθl)12[k||2kmL(wdwm)2zdzm+kmL(zdzm+zmzd)]sinθmsinθl}[kmLsinθlk||(wdwm)zmsinθm]1,
(2)
where for convenience we introduced the dimensionless parameters
θb=kd,θd=kda,θm=kmTb,θl=kmLb,
(3a)
zd=kdk0εd,wd=k||k0,zm=kmTk0εmT,wd=k||k0εmT,
(3b)
and where k⊥ represents the Bloch wavevector in the direction of the periodicity, k|| the wavevector along the layers, and k0 = ω/c the free-space wavevector. We also introduced the derived wavevectors k,d=k02εdk||2, kmT=k02εmTk||2, and kmL=(ω2+iγωωp2)/β2k||2.

We note that solving the hydrodynamic Drude model requires boundary conditions in addition to the usual Maxwell boundary conditions. The dispersion relation Eq. (2) was derived using the continuity of the normal component of the free-electron current as the additional boundary condition. Details can be found in Refs. [26

26. S. Raza, G. Toscano, A.-P. Jauho, M. Wubs, and N. A. Mortensen, “Unusual resonances in nanoplasmonic structures due to nonlocal response,” Phys. Rev. B 84, 121412(R) (2011) [CrossRef] .

, 33

33. W. Yan, M. Wubs, and N. A. Mortensen, “Hyperbolic metamaterials: nonlocal response regularizes broadband supersingularity,” Phys. Rev. B 86, 205429 (2012) [CrossRef] .

, 34

34. P. Jewsbury, “Electrodynamic boundary conditions at metal interfaces,” J. Phys. F: Met. Phys. 11, 195–206 (1981) [CrossRef] .

]. The exact dispersion relation for nonlocal response Eq. (2) reduces to the exact dispersion relation in the LRA by putting the longitudinal permittivity εmL(k,ω) of Eq. (1) equal to the transverse permittivity εmT(k,ω) of Eq. (1), in other words by taking the nonlocal parameter β to zero.

3. Effective nonlocal material parameters

As stated before, we consider sub-wavelength unit cells much smaller than optical wavelengths, but with metal layers much larger than the wavelengths of their longitudinal pressure waves. So we focus on the situation where the following two parameters are small,
k0d1,1|kmLb|1,
(4)
where kmL=ω2+iγωωp2/β is the longitudinal wavevector. Furthermore, we will only consider the frequency range ω < ωp, where εmT<0 and the dispersion curve of the HMM could be a hyperbola in the LRA. We now make a first-order Taylor approximation in the small parameters of Eq. (4) to the exact dispersion relation Eq. (2), and obtain the approximate dispersion relation
k2=k02ε||lock||2ε||locεloc{k||2ε||locεhdm+Δla}.
(5)
The two terms in the brace are two leading correction terms that originate from the nonlocal response of the free electrons and from the finite size of the unit cell, respectively. Without them, i.e. when neglecting both the finiteness of the unit cells and nonlocal response, we have the well-known dispersion relation in the LRA
k2=k02ε||lock||2ε||locεloc,
(6)
where ε||loc and εloc are the effective local permittivities [9

9. B. Wood, J. Pendry, and D. Tsai, “Directed subwavelength imaging using a layered metal-dielectric system,” Phys. Rev. B 74, 115116 (2006) [CrossRef] .

]
ε||loc=fdεd+fmεm,εloc=1fdεd1+fmεm1,
(7)
where fd = a/d and fm = b/d. We note that Eq. (7) is satisfied when the unit-cell thickness is much smaller than the free-space wavelength indicated by the first inequality in Eq. (4). Obviously, when ε||locεloc<0, Eq. (6) describes a hyperbolic dispersion curve. However, we are now rather interested in nonlocal response effects in realistic HMMs, so we investigate the importance of the terms in the brace in Eq. (5), which are given in terms of
εhdm=kmLd2iεmTεmT1and
(8a)
Δla=112εdεmTfdfmd2(k02k||2εloc)2+112ε||locd2[2kd2(kmT)2fmfdε||locεdεmT+(kd)4fd3εd+(kmT)4fm3εmT].
(8b)
From the expression (8a) for εhdm, it follows that the local-response limit is found in the limit kmLd rather than by taking the size d of the unit cell to zero, see also Ref. [33

33. W. Yan, M. Wubs, and N. A. Mortensen, “Hyperbolic metamaterials: nonlocal response regularizes broadband supersingularity,” Phys. Rev. B 86, 205429 (2012) [CrossRef] .

] on this point. In the following we will keep the first (i.e. the nonlocal) correction term, while neglecting the other correction term Δla. The sole justification for doing so is that the second term turns out to be negligible in comparison to the first one for the HMMs that we consider, as we illustrate numerically below. So, as an improved approximation to Eq. (6), we keep the correction term concerning εhdm, and then obtain the approximate nonlocal dispersion relation that is central to our present study,
k2=k02ε||lock||2ε||locεnloc,
(9)
in terms of the new effective permittivity in the direction of the periodicity corrected by the nonlocal response
1εnloc=1εloc+1εhdm.
(10)
The other tensor component ε||loc of the dielectric tensor has no such nonlocal correction.

Since 1/|kmLd|1, we usually have εlocεhdm, and then εnlocεloc is a good approximation. However, for large εloc and especially in the extreme case that εloc, it may occur that εloc becomes much larger than εhdm, so that εnlocεhdm by virtue of Eq. (10). So in this case, the nonlocal response has the important role of replacing the infinite εloc by the finite εhdm in the dispersion relation. But when does it occur that εloc? This immediately follows from Eq. (7), and occurs when fdεT + fmεd vanishes. In particular, neglecting the Drude damping γ of the metal, the frequency ωresloc for which εloc is given by
ωresloc=ωpfdfd+fmεd.
(11)
Let us also consider another extreme case, namely εnloc. This is achieved when εloc=εhdm, which happens at the frequency
ωresnlocωresloc(1+εdkmLa).
(12)
At this frequency ωresnloc, the nonlocal response changes the finite εloc into the infinite εnloc, again a strong nonlocal effect.

To give a physical explanation on nonlocal blue-shift independent of the metal layer thickness, we simplify the discussion by considering the dielectric layer being free space with εd = 1. From Eq. (11), we have the local resonance frequency ωresloc=ωpa/d. With nonlocal response, the surface charge at the metal boundary is smeared out. Effectively, we could view the surface charge as being displaced into the metal layer by a distance of 1/kmL. Accordingly, the effective thickness of the free-space layer is increased to aeff=a+2/kmL, where the factor 2 comes from the two interface boundaries of the metal layer. Taking aeff into Eq. (11), we have ωresnlocωresloc(1+1/(kmLa)), which is just Eq. (12) with εd = 1, and independent of the metal layer thickness.

4. Effects of nonlocal response on the dispersion curve: numerical analysis

To numerically illustrate the effects of the nonlocal response on the HMM, we choose a specific example of the HMM with a = 6nm, b = 3nm, and εd = 10. We choose the metal to be Au, and describe it by only its free-electron response, with parameters h̄ωp = 8.812eV, h̄γ = 0.0752eV, and vF = 1.39 × 106m/s. Here, we note that it is a tough task to fabricate the HMM with a unit cell of a 3nm-thick metal layer and a dielectric layer of 6nm. However, with further progress in nano-fabrication techniques, we believe that such a HMM structure will be practically feasible in the near future. For example, very recently experimentalists have succeeded in fabricating a 3nm-thick Ag film by depositing the film on a 1nm copper seed layer [39

39. N. Formica, D. S. Ghosh, A. Carrilero, T. L. Chen, R. E. Simpson, and V. Pruneri, “Ultrastable and Atomically Smooth Ultrathin Silver Films Grown on a Copper Seed Layer,” ACS Appl. Mater. Interfaces 5, 3048–3053 (2013) [CrossRef] [PubMed] .

].

Fig. 1 (a) k0d and 1/|kmLb|, (b) real part of ε||loc, and (c) real parts of εloc and εnloc, of the HMM. The unit cell of the HMM shown in the inset of (a) has a = 6nm, b = 3nm, εd = 10, and the yellow metal layer is Au.

Figure 2(a) and (b) illustrate the dispersion curves at ω = 0.1ωp and ω = 0.6ωp, respectively. These frequencies are far away from the resonances ωresloc and ωresnloc. The small damping loss is neglected for a clearer illustration of the dispersion curve. Several observations can be extracted from Fig. 2. Firstly, the exact local and nonlocal dispersion curves agree well with those based on the effective material parameters, which confirms the validity of the approximate theoretical results of Sec. 3. Secondly, the local and nonlocal curves only show a slight difference, agreeing well with Fig. 1(c) where we see εnlocεloc both for ω = 0.1 ωp and 0.6ωp.

Fig. 2 Dispersion curves of the HMM at (a) ω = 0.1ωp and (b) ω = 0.6ωp. The HMM is as that in Fig. 1 except that the loss is neglected. Red solid curves for the exact local dispersion, blue solid curves for the exact nonlocal dispersion, red dashed curves for the approximated local dispersion of Eq. (6), and blue dashed curves for the approximated nonlocal dispersion of Eq. (9).

Figure 3(a) and (b) show the on-resonance dispersion curves, namely for ω=ωresloc=0.41ωp = and for ω=ωresnloc=0.465ωp, respectively. Loss is again neglected, as in Fig. 2. At both frequencies, the nonlocal response modifies the dispersion curve noticeably. In particular, at ω = 0.41ωp, the local dispersion curve consists of two nearly flat lines of k=±k0ε||loc. With the nonlocal response, however, the dispersion curve becomes a closed ellipse! This remarkable difference can be understood from the difference between εloc and εnloc. In particular, εloc diverges while εnloc=41 stays finite in the lossless case. When including the loss, the effective parameters change to εloc=200+500i and εnloc=43.7+3i. The loss would only slightly modify the dispersion curves of Fig. 3(a). In Fig. 3(b) we display dispersion relations at ω = 0.465ωp. Here the local dispersion curve is a hyperbola. With the nonlocal response, however, the dispersion curve becomes nearly flat lines, which can be attributed to the extremely large value of εnloc at this frequency ωresnloc.

Fig. 3 Dispersion curves of the HMM at (a) ω=ωresloc=0.41ωp and (b) ω=ωresnloc=0.465ωp. The HMM is as that in Fig. 1 except that the loss is neglected. Red solid curves denote the exact local dispersion, blue solid curves the exact nonlocal dispersion; red dashed curves for the approximated local dispersion curves of Eq. (6), and blue dashed curves for the approximated dispersion curves of Eq. (9).

5. Effects of nonlocal response on a hyperbolic metamaterial lens

A HMM slab can operate as a superlens with subwavelength resolution, since the hyperbolic dispersion curve supports propagating waves with arbitrarily high wavevectors, which can transfer the evanescent information of the object. In the LRA, it is known that the HMM with its flat dispersion curve at ωresloc [recall Fig. 3(a)] is especially favorable for subwavelength imaging [8

8. P. Belov and Y. Hao, “Subwavelength imaging at optical frequencies using a transmission device formed by a periodic layered metal-dielectric structure operating in the canalization regime,” Phys. Rev. B 73, 113110 (2006) [CrossRef] .

, 12

12. Z. Jacob, L. Alekseyev, and E. Narimanov, “Optical hyperlens: Far-field imaging beyond the diffraction limit,” Opt. Express 14, 8427–8256 (2006) [CrossRef] .

]. The reason is that the flat dispersion curve with the constant k ensures that all plane-wave components experience the same phase changes after transmission through the HMM slab, at least if reflections can be neglected. Actually, the reflections can be suppressed by appropriate choice of the thickness l of the HMM lens, and even be made to vanish by choosing kl = , where n is a positive integer. In theory this could lead to a perfect image at x = 0.

Let us now investigate how nonlocal response may influence the subwavelength imaging characteristics of the HMM lens. As demonstrated in the above sections, the nonlocal response sets the infinite εloc to a finite value and accordingly destroys the desired flat dispersion curve at the frequency ωresloc where one would normally choose to operate for perfect imaging, based on the LRA. Nevertheless, at a blueshifted frequency ωresnloc, the local effective dielectric function εloc may be finite, but the hydrodynamic Drude model predicts instead that εnloc diverges, with the concomitant flat dispersion curve suitable for subwavelength imaging. Thus nonlocal response is expected to strongly affect the performance of HMM superlenses for frequencies ω around ωresloc and ωresnloc.

As an example, we consider a HMM slab with l = 36d, i.e. composed of 36 unit cells, in a free-space background with the two boundaries at x = −l and x = 0. The unit cell is as in Fig. 1, and is arranged in a symmetric sandwich structure with the metal layer at the center. A line dipole source is positioned to the left of the HMM slab, with x-coordinate −lxs and y-coordinate 0, and is represented by J = δ(x + l + xs)δ(y)ŷ. We choose the distance to the HMM slab to be xs = 10nm. The interaction between the current source and the HMM slab is solved by using the transfer-matrix method of Refs. [23

23. W. L. Mochán, M. Castillo-Mussot, and R. G. Barrera, “Effect of plasma waves on the optical properties of metal-insulator superlattices,” Phys. Rev. B 15, 1088–1098 (1987) [CrossRef] .

, 33

33. W. Yan, M. Wubs, and N. A. Mortensen, “Hyperbolic metamaterials: nonlocal response regularizes broadband supersingularity,” Phys. Rev. B 86, 205429 (2012) [CrossRef] .

].

Figure 4(a1) and (b1) demonstrate the transmitted electric-field intensity at ω=ωresloc=0.41ωp for local and nonlocal response, respectively. Metal loss is taken into account. The spatial coordinates in Fig. 4(a1) and (b1) are normalized by the free-space wavelength λp that corresponds to the plasma frequency. In the local case, we have the known the dispersion of Fig. 3(a) featuring flat lines. There kl approximately equals ≈ 4.25π, indicating that reflections do exist. However, the transmission coefficients for the different wave components vary more smoothly as a function of k||, thanks to the flat dispersion curve with the constant k. Accordingly, after propagation through the HMM from x = −l to x = 0, an image is formed near x = 0, as seen in Fig. 4(a1). The size of the image is approximately λp/2, i.e., 0.2λ0 with λ0 being the operating wavelength corresponding to ω0 = 0.41ωp, so that the size of the image is in the subwavelength scale. By contrast, in the nonlocal case the dispersion curve turns into an ellipse. This closed dispersion curve sets an upper wavevector cutoff for the evanescent waves, and the wave components below the cutoff experience different phase changes. Accordingly, the quality of the image becomes worse, see Fig. 4(a2).

Fig. 4 Transmitted electric-field intensity distribution for a line dipole source J = δ(x + l + xs)δ(y)ŷ positioned to the left of a HMM slab that has its left interface at x = −l and right interface at x = 0. For ω=0.41ωpωresloc, panel (a1) shows the intensity for local response, (b1) for nonlocal response, and (c1) the intensity along y with x = 0. Panels (a2), (b2), and (c2) are the analogous graphs for ω=0.465ωpωresnloc. In panels (c1) and (c2), the green dashed curves are for the case without the HMM slab, the red solid curves for the HMM with local response, and the blue solid curves for the HMM with nonlocal response. The HMM unit cell is as that in Fig. 1, xs = 10nm and l = 36d. The spatial coordinates in all figures are normalized by the free space wavelength at plasma frequency denoted as λp.

In Fig. 4(c1), we depict the electric-field intensity again at ωresloc as a function of y at the HMM boundary x = 0, and also for the case without the HMM slab. It is seen that the electric-field intensity is nearly vanishing in the absence of the HMM slab, since the evanescent components vanish after traveling a distance of l, which is of the order of one free-space wavelength. With the HMM slab in place, the electric-field intensity in the local case shows a subwavelength image that peaks at y = 0, with a full width at half maximum (FWHM) of only 0.42λp. With nonlocal response, however, the electric-field intensity distribution for ωresloc becomes flatter, with a double rather than a single peak, with peak intensities at y = ±0.55λp.

Let us now turn to the other resonance frequency, namely ω=ωresnloc of Eq. (12). Figure 4(a2) and (b2) show the transmitted electric-field intensity for the local and nonlocal cases, respectively, at ω=ωresnloc=0.465ωp. At this frequency, the local dispersion curve is a hyperbola, while the nonlocal dispersion shows two flat lines. In the nonlocal case, we have kl ≈ 5π, and this value of nearly an integer times π indicates that reflections are nearly zero. As a result, the hydrodynamic Drude model predicts a better focusing performance of the HMM slab at ω=0.465ωp=ωresnloc than does the local-response theory, compare Fig. 4(a2) and (b2). In Fig. 4(c2), the electric-field intensity along y at x = 0 is shown. In the local case, the electric-field intensity shows several peaks with the strongest two at y = ±0.33λp. In the nonlocal case, the electric-field intensity is peaked at y = 0 with a subwavelength FWHM of only 0.4λp.

6. Detecting nonlocal response by near-field measurement

Figure 4 indicates the possibility of detecting the nonlocal response experimentally by measuring the transmitted near-field distribution at the surface of a HMM superlens, which in our setup would be its right interface at x = 0. The images in Fig. 4(c1,c2) correspond to hypothetical measurements with infinitely small detectors. In experiments, the measured near-field signal will rather be an area-averaged electric-field intensity
Iav1DDd2r|E(r)|2,
(13)
where D is the finite detection area of the detector. Let us now assume that we have a near-field detector with detection area in the yz-plane, with a square shape of size 40nm × 40nm, touching the HMM interface at x = 0. For the same light source interacting with the HMM slab as in Fig. 4, we depict in Fig. 5 the calculated Iav as a function of the y-coordinate of the center of the detector. It is seen that local and nonlocal response models give significantly different predictions for the measured signal, and thus the differences between the two models survive detection area averaging. The single peaks at 0.41ωp for local response and at 0.465ωp for nonlocal response are naturally broader than in Fig. 4(c1,c2), also due to the area averaging. Interestingly, as the frequency increases from 0.41ωp to 0.465ωp, the distribution of Iav becomes broader in the LRA, but narrower in the hydrodynamic Drude model.

Fig. 5 Calculated detector-area-averaged near-field intensity Iav of the light emitted by the same source as in Fig. 4 and transmitted through the same HMM slab. The detector is square-shaped with size 40nm×40nm, touching the HMM right interface at x = 0. It scans along the y-direction, and the y-coordinate on the horizontal axes in panels (a) for the LRA and (b) for the HDM corresponds to that of the center of the detector.

7. Conclusions

We investigated the effects of the hydrodynamic nonlocal response on hyperbolic metamaterials with periodicity in the subwavelength regime of the transverse optical wave, but much larger than the wavelength of the longitudinal hydrodynamic pressure waves. It is found that the nonlocal response corrects the effective permittivity tensor element εloc in the periodicity direction to the new form εnloc of Eq. (10). Around the frequencies ωresloc and ωresnloc corresponding to εloc and εnloc, respectively, εloc and εnloc show noticeable differences, even leading to completely different dispersion curves with and without the nonlocal response.

We find that nonlocal response blueshifts the resonance frequency of HMMs from ωresloc [Eq. (11)] to ωresnloc [Eq. (12)]. The relative blueshift has an interesting simple form independent of the thickness of the metal layers. Similar nonlocal blueshifts for single nanoplasmonic particles have been predicted before, and significant blueshifts have also been measured [27

27. J. A. Scholl, A. L. Koh, and J. A. Dionne, “Quantum plasmon resonances of individual metallic nanoparticles,” Nature 483, 421–427 (2012) [CrossRef] [PubMed] .

, 29

29. S. Raza, N. Stenger, S. Kadkhodazadeh, S. V. Fischer, N. Kostesha, A.-P. Jauho, A. Burrows, M. Wubs, and N. A. Mortensen, “Blueshift of the surface plasmon resonance in silver nanoparticles studied with EELS,” Nanophotonics 2, 131 (2013) [CrossRef] .

]; how much of these can be attributed to hydrodynamic effects is a hot topic [27

27. J. A. Scholl, A. L. Koh, and J. A. Dionne, “Quantum plasmon resonances of individual metallic nanoparticles,” Nature 483, 421–427 (2012) [CrossRef] [PubMed] .

, 29

29. S. Raza, N. Stenger, S. Kadkhodazadeh, S. V. Fischer, N. Kostesha, A.-P. Jauho, A. Burrows, M. Wubs, and N. A. Mortensen, “Blueshift of the surface plasmon resonance in silver nanoparticles studied with EELS,” Nanophotonics 2, 131 (2013) [CrossRef] .

, 35

35. R. C. Monreal, T. J. Antosiewicz, and S. P. Apell, “Plasmons do not go that quantum,” arXiv:1304.3023 (2013).

]. Similar discussions apply to nanowire dimers [36

36. L. Stella, P. Zhang, F. J. García-Vidal, A. Rubio, and P. García-González, “Performance of nonlocal optics when applied to plasmonic nanostructures,” J. Phys. Chem. C 117, 8941–8949 (2013) [CrossRef] .

38

38. K. Andersen, K. L. Jensen, and K. S. Thygesen, “Hybridization of quantum plasmon modes in coupled nanowires: From the classical to the tunneling regime,” arXiv:1304.4754 (2013).

].

Furthermore, we predict that nonlocal response shows its mark in the performance of a finite HMM slab as a focusing lens: when increasing the operating frequency from ωresloc to ωresnloc, for local response the near-field distribution of the transmitted light shows an image that gets out of focus, whereas the focus would improve instead according to the nonlocal-response theory. We propose to test the blueshift and these contrary predictions experimentally, as a clear and interesting test whether nonlocal response can be observed in hyperbolic metamaterials.

Acknowledgments

References and links

1.

D. R. Smith and D. Schurig, “Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors,” Phys. Rev. Lett. 90, 077405 (2003) [CrossRef] [PubMed] .

2.

I. I. Smolyaninov, “Vacuum in a Strong Magnetic Field as a Hyperbolic Metamaterial,” Phys. Rev. Lett. 107, 253903 (2011) [CrossRef] .

3.

H. N. S. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological transitions in metamaterials,” Science 336, 205–209 (2012) [CrossRef] [PubMed] .

4.

Z. Jacob, I. Smolyaninov, and E. Narimanov, “Broadband Purcell effect: Radiative decay engineering with metamaterials,” Appl. Phys. Lett. 100, 181105 (2012).

5.

A. N. Poddubny, P. A. Belov, G. V. Naik, and Y. S. Kivshar, “Spontaneous radiation of a finite-size dipole emitter in hyperbolic media,” Phys. Rev. A 84, 023807 (2011) [CrossRef] .

6.

T. Tumkur, G. Zhu, P. Black, Y. A. Barnakov, C. E. Bonner, and M. A. Noginov, “Control of spontaneous emission in a volume of functionalized hyperbolic metamaterial,” Appl. Phys. Lett. 99, 151115 (2011) [CrossRef] .

7.

M. A. Noginov, Y. A. Barnakov, G. Zhu, T. Tumkur, H. Li, and E. E. Narimanov, “Bulk photonic metamaterial with hyperbolic dispersion,” Appl. Phys. Lett. 94, 151105 (2009) [CrossRef] .

8.

P. Belov and Y. Hao, “Subwavelength imaging at optical frequencies using a transmission device formed by a periodic layered metal-dielectric structure operating in the canalization regime,” Phys. Rev. B 73, 113110 (2006) [CrossRef] .

9.

B. Wood, J. Pendry, and D. Tsai, “Directed subwavelength imaging using a layered metal-dielectric system,” Phys. Rev. B 74, 115116 (2006) [CrossRef] .

10.

X. Li, S. He, and Y. Jin, “Subwavelength focusing with a multilayered Fabry-Perot structure at optical frequencies,” Phys. Rev. B 75, 045103 (2007) [CrossRef] .

11.

A. Salandrino and N. Engheta, “Far-field subdiffraction optical microscopy using metamaterial crystals: Theory and simulations,” Phys. Rev. B 74, 075103 (2006) [CrossRef] .

12.

Z. Jacob, L. Alekseyev, and E. Narimanov, “Optical hyperlens: Far-field imaging beyond the diffraction limit,” Opt. Express 14, 8427–8256 (2006) [CrossRef] .

13.

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315, 1686 (2007) [CrossRef] [PubMed] .

14.

M. Yan and N. A. Mortensen, “Hollow-core infrared fiber incorporating metal-wire metamaterial,” Opt. Express 17, 14851–14864 (2009) [CrossRef] [PubMed] .

15.

C. L. Cortes, W. Newman, S. Molesky, and Z. Jacob, “Quantum nanophotonics using hyperbolic metamaterials,” J. Opt. 14, 063001 (2012) [CrossRef] .

16.

J. Elser, V. Podolksiy, I. Salakhutdinov, and I. Avrutsky, “Nonlocal effects in effective-medium response of nanolayered metamaterials,” Appl. Phys. Lett. 90, 191109 (2007).

17.

A. Chebykin, A. Orlov, A. Vozianova, S. Maslovski, Y. S. Kivshar, and P. Belov, “Nonlocal effective medium model for multilayered metal-dielectric metamaterials,” Phys. Rev. B 84, 115438 (2011) [CrossRef] .

18.

A. Chebykin, A. Orlov, C. Simovski, Y. S. Kivshar, and P. Belov, “Nonlocal effective parameters of multilayered metal-dielectric metamaterials,” Phys. Rev. B 86, 115420 (2012) [CrossRef] .

19.

L. Shen, T. Yang, and Y. Chau, “Effect of internal period on the optical dispersion of indefinite-medium materials,” Phys. Rev. B 77, 205124 (2008) [CrossRef] .

20.

L. Shen, T. Yang, and Y. Chau, “50/50 beam splitter using a one-dimensional metal photonic crystal with parabo-lalike dispersion,” Appl. Phys. Lett. 90, 251909 (2007) [CrossRef] .

21.

F. Bloch, “Bremsvermögen von Atomen mit mehreren Elektronen,” Z. Phys. A 81, 363–376 (1933).

22.

A. D. Boardman, Electromagnetic Surface Modes (John Wiley and Sons, Chichester, 1982).

23.

W. L. Mochán, M. Castillo-Mussot, and R. G. Barrera, “Effect of plasma waves on the optical properties of metal-insulator superlattices,” Phys. Rev. B 15, 1088–1098 (1987) [CrossRef] .

24.

C. David and F. J. García de Abajo, “Spatial nonlocality in the optical response of metal nanoparticles,” J. Phys. Chem. C 115, 19470–19475 (2011) [CrossRef] .

25.

F. J. García de Abajo, “Nonlocal effects in the plasmons of strongly interacting nanoparticles, dimers, and waveguides,” J. Phys. Chem. C 112, 17983–17987 (2008) [CrossRef] .

26.

S. Raza, G. Toscano, A.-P. Jauho, M. Wubs, and N. A. Mortensen, “Unusual resonances in nanoplasmonic structures due to nonlocal response,” Phys. Rev. B 84, 121412(R) (2011) [CrossRef] .

27.

J. A. Scholl, A. L. Koh, and J. A. Dionne, “Quantum plasmon resonances of individual metallic nanoparticles,” Nature 483, 421–427 (2012) [CrossRef] [PubMed] .

28.

C. Ciracì, R. T. Hill, J. J. Mock, Y. Urzhumov, A. I. Fernández-Domínguez, S. A. Maier, J. B. Pendry, A. Chilkoti, and D. R. Smith, “Probing the ultimate limits of plasmonic enhancement,” Science 337, 1072–1074 (2012) [CrossRef] [PubMed] .

29.

S. Raza, N. Stenger, S. Kadkhodazadeh, S. V. Fischer, N. Kostesha, A.-P. Jauho, A. Burrows, M. Wubs, and N. A. Mortensen, “Blueshift of the surface plasmon resonance in silver nanoparticles studied with EELS,” Nanophotonics 2, 131 (2013) [CrossRef] .

30.

G. Toscano, S. Raza, A.-P. Jauho, M. Wubs, and N. A. Mortensen, “Modified field enhancement and extinction in plasmonic nanowire dimers due to nonlocal response,” Opt. Express 13, 4176–4188 (2012) [CrossRef] .

31.

G. Toscano, S. Raza, S. Xiao, M. Wubs, A.-P. Jauho, S. I. Bozhevolnyi, and N. A. Mortensen, “Surface-enhanced Raman spectroscopy (SERS): nonlocal limitations,” Opt. Lett. 37, 2538–2540 (2012) [CrossRef] [PubMed] .

32.

A. I. Fernández-Domínguez, A. Wiener, F. J. García-Vidal, S. A. Maier, and J. B. Pendry, “Transformation-optics description of nonlocal effects in plasmonic nanostructures,” Phys. Rev. Lett. 108, 106802 (2012) [CrossRef] [PubMed] .

33.

W. Yan, M. Wubs, and N. A. Mortensen, “Hyperbolic metamaterials: nonlocal response regularizes broadband supersingularity,” Phys. Rev. B 86, 205429 (2012) [CrossRef] .

34.

P. Jewsbury, “Electrodynamic boundary conditions at metal interfaces,” J. Phys. F: Met. Phys. 11, 195–206 (1981) [CrossRef] .

35.

R. C. Monreal, T. J. Antosiewicz, and S. P. Apell, “Plasmons do not go that quantum,” arXiv:1304.3023 (2013).

36.

L. Stella, P. Zhang, F. J. García-Vidal, A. Rubio, and P. García-González, “Performance of nonlocal optics when applied to plasmonic nanostructures,” J. Phys. Chem. C 117, 8941–8949 (2013) [CrossRef] .

37.

T. Teperik, P. Nordlander, J. Aizpurua, and A. G. Borisov, “Quantum plasmonics: Nonlocal effects in coupled nanowire dimer,” arXiv:1302.3339 (2013).

38.

K. Andersen, K. L. Jensen, and K. S. Thygesen, “Hybridization of quantum plasmon modes in coupled nanowires: From the classical to the tunneling regime,” arXiv:1304.4754 (2013).

39.

N. Formica, D. S. Ghosh, A. Carrilero, T. L. Chen, R. E. Simpson, and V. Pruneri, “Ultrastable and Atomically Smooth Ultrathin Silver Films Grown on a Copper Seed Layer,” ACS Appl. Mater. Interfaces 5, 3048–3053 (2013) [CrossRef] [PubMed] .

ToC Category:
Metamaterials

History
Original Manuscript: April 18, 2013
Revised Manuscript: May 14, 2013
Manuscript Accepted: May 15, 2013
Published: June 17, 2013

Virtual Issues
Hyperbolic Metamaterials (2013) Optics Express

Citation
Wei Yan, N. Asger Mortensen, and Martijn Wubs, "Hyperbolic metamaterial lens with hydrodynamic nonlocal response," Opt. Express 21, 15026-15036 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-12-15026


Sort:  Author  |  Year  |  Journal  |  Reset  

References

  1. D. R. Smith and D. Schurig, “Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors,” Phys. Rev. Lett.90, 077405 (2003). [CrossRef] [PubMed]
  2. I. I. Smolyaninov, “Vacuum in a Strong Magnetic Field as a Hyperbolic Metamaterial,” Phys. Rev. Lett.107, 253903 (2011). [CrossRef]
  3. H. N. S. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological transitions in metamaterials,” Science336, 205–209 (2012). [CrossRef] [PubMed]
  4. Z. Jacob, I. Smolyaninov, and E. Narimanov, “Broadband Purcell effect: Radiative decay engineering with metamaterials,” Appl. Phys. Lett.100, 181105 (2012).
  5. A. N. Poddubny, P. A. Belov, G. V. Naik, and Y. S. Kivshar, “Spontaneous radiation of a finite-size dipole emitter in hyperbolic media,” Phys. Rev. A84, 023807 (2011). [CrossRef]
  6. T. Tumkur, G. Zhu, P. Black, Y. A. Barnakov, C. E. Bonner, and M. A. Noginov, “Control of spontaneous emission in a volume of functionalized hyperbolic metamaterial,” Appl. Phys. Lett.99, 151115 (2011). [CrossRef]
  7. M. A. Noginov, Y. A. Barnakov, G. Zhu, T. Tumkur, H. Li, and E. E. Narimanov, “Bulk photonic metamaterial with hyperbolic dispersion,” Appl. Phys. Lett.94, 151105 (2009). [CrossRef]
  8. P. Belov and Y. Hao, “Subwavelength imaging at optical frequencies using a transmission device formed by a periodic layered metal-dielectric structure operating in the canalization regime,” Phys. Rev. B73, 113110 (2006). [CrossRef]
  9. B. Wood, J. Pendry, and D. Tsai, “Directed subwavelength imaging using a layered metal-dielectric system,” Phys. Rev. B74, 115116 (2006). [CrossRef]
  10. X. Li, S. He, and Y. Jin, “Subwavelength focusing with a multilayered Fabry-Perot structure at optical frequencies,” Phys. Rev. B75, 045103 (2007). [CrossRef]
  11. A. Salandrino and N. Engheta, “Far-field subdiffraction optical microscopy using metamaterial crystals: Theory and simulations,” Phys. Rev. B74, 075103 (2006). [CrossRef]
  12. Z. Jacob, L. Alekseyev, and E. Narimanov, “Optical hyperlens: Far-field imaging beyond the diffraction limit,” Opt. Express14, 8427–8256 (2006). [CrossRef]
  13. Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science315, 1686 (2007). [CrossRef] [PubMed]
  14. M. Yan and N. A. Mortensen, “Hollow-core infrared fiber incorporating metal-wire metamaterial,” Opt. Express17, 14851–14864 (2009). [CrossRef] [PubMed]
  15. C. L. Cortes, W. Newman, S. Molesky, and Z. Jacob, “Quantum nanophotonics using hyperbolic metamaterials,” J. Opt.14, 063001 (2012). [CrossRef]
  16. J. Elser, V. Podolksiy, I. Salakhutdinov, and I. Avrutsky, “Nonlocal effects in effective-medium response of nanolayered metamaterials,” Appl. Phys. Lett.90, 191109 (2007).
  17. A. Chebykin, A. Orlov, A. Vozianova, S. Maslovski, Y. S. Kivshar, and P. Belov, “Nonlocal effective medium model for multilayered metal-dielectric metamaterials,” Phys. Rev. B84, 115438 (2011). [CrossRef]
  18. A. Chebykin, A. Orlov, C. Simovski, Y. S. Kivshar, and P. Belov, “Nonlocal effective parameters of multilayered metal-dielectric metamaterials,” Phys. Rev. B86, 115420 (2012). [CrossRef]
  19. L. Shen, T. Yang, and Y. Chau, “Effect of internal period on the optical dispersion of indefinite-medium materials,” Phys. Rev. B77, 205124 (2008). [CrossRef]
  20. L. Shen, T. Yang, and Y. Chau, “50/50 beam splitter using a one-dimensional metal photonic crystal with parabo-lalike dispersion,” Appl. Phys. Lett.90, 251909 (2007). [CrossRef]
  21. F. Bloch, “Bremsvermögen von Atomen mit mehreren Elektronen,” Z. Phys. A81, 363–376 (1933).
  22. A. D. Boardman, Electromagnetic Surface Modes (John Wiley and Sons, Chichester, 1982).
  23. W. L. Mochán, M. Castillo-Mussot, and R. G. Barrera, “Effect of plasma waves on the optical properties of metal-insulator superlattices,” Phys. Rev. B15, 1088–1098 (1987). [CrossRef]
  24. C. David and F. J. García de Abajo, “Spatial nonlocality in the optical response of metal nanoparticles,” J. Phys. Chem. C115, 19470–19475 (2011). [CrossRef]
  25. F. J. García de Abajo, “Nonlocal effects in the plasmons of strongly interacting nanoparticles, dimers, and waveguides,” J. Phys. Chem. C112, 17983–17987 (2008). [CrossRef]
  26. S. Raza, G. Toscano, A.-P. Jauho, M. Wubs, and N. A. Mortensen, “Unusual resonances in nanoplasmonic structures due to nonlocal response,” Phys. Rev. B84, 121412(R) (2011). [CrossRef]
  27. J. A. Scholl, A. L. Koh, and J. A. Dionne, “Quantum plasmon resonances of individual metallic nanoparticles,” Nature483, 421–427 (2012). [CrossRef] [PubMed]
  28. C. Ciracì, R. T. Hill, J. J. Mock, Y. Urzhumov, A. I. Fernández-Domínguez, S. A. Maier, J. B. Pendry, A. Chilkoti, and D. R. Smith, “Probing the ultimate limits of plasmonic enhancement,” Science337, 1072–1074 (2012). [CrossRef] [PubMed]
  29. S. Raza, N. Stenger, S. Kadkhodazadeh, S. V. Fischer, N. Kostesha, A.-P. Jauho, A. Burrows, M. Wubs, and N. A. Mortensen, “Blueshift of the surface plasmon resonance in silver nanoparticles studied with EELS,” Nanophotonics2, 131 (2013). [CrossRef]
  30. G. Toscano, S. Raza, A.-P. Jauho, M. Wubs, and N. A. Mortensen, “Modified field enhancement and extinction in plasmonic nanowire dimers due to nonlocal response,” Opt. Express13, 4176–4188 (2012). [CrossRef]
  31. G. Toscano, S. Raza, S. Xiao, M. Wubs, A.-P. Jauho, S. I. Bozhevolnyi, and N. A. Mortensen, “Surface-enhanced Raman spectroscopy (SERS): nonlocal limitations,” Opt. Lett.37, 2538–2540 (2012). [CrossRef] [PubMed]
  32. A. I. Fernández-Domínguez, A. Wiener, F. J. García-Vidal, S. A. Maier, and J. B. Pendry, “Transformation-optics description of nonlocal effects in plasmonic nanostructures,” Phys. Rev. Lett.108, 106802 (2012). [CrossRef] [PubMed]
  33. W. Yan, M. Wubs, and N. A. Mortensen, “Hyperbolic metamaterials: nonlocal response regularizes broadband supersingularity,” Phys. Rev. B86, 205429 (2012). [CrossRef]
  34. P. Jewsbury, “Electrodynamic boundary conditions at metal interfaces,” J. Phys. F: Met. Phys.11, 195–206 (1981). [CrossRef]
  35. R. C. Monreal, T. J. Antosiewicz, and S. P. Apell, “Plasmons do not go that quantum,” arXiv:1304.3023 (2013).
  36. L. Stella, P. Zhang, F. J. García-Vidal, A. Rubio, and P. García-González, “Performance of nonlocal optics when applied to plasmonic nanostructures,” J. Phys. Chem. C117, 8941–8949 (2013). [CrossRef]
  37. T. Teperik, P. Nordlander, J. Aizpurua, and A. G. Borisov, “Quantum plasmonics: Nonlocal effects in coupled nanowire dimer,” arXiv:1302.3339 (2013).
  38. K. Andersen, K. L. Jensen, and K. S. Thygesen, “Hybridization of quantum plasmon modes in coupled nanowires: From the classical to the tunneling regime,” arXiv:1304.4754 (2013).
  39. N. Formica, D. S. Ghosh, A. Carrilero, T. L. Chen, R. E. Simpson, and V. Pruneri, “Ultrastable and Atomically Smooth Ultrathin Silver Films Grown on a Copper Seed Layer,” ACS Appl. Mater. Interfaces5, 3048–3053 (2013). [CrossRef] [PubMed]

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

Figures

Fig. 1 Fig. 2 Fig. 3
 
Fig. 4 Fig. 5
 

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited