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This leads to numerous applications, such #189117 - $15.00 USD Received 18 Apr 2013; revised 14 May 2013; accepted 15 May 2013; published 17 Jun 2013 (C) 2013 OSA 17 June 2013 | Vol. 21, No. 12 | DOI:10.1364/OE.21.015026 | OPTICS EXPRESS 15027
as enhanced-light interactions [4–7], subwavelength imaging [8–13], negative refraction [1], and low-loss fiber cladding [14]. HMMs are now also be emphasized for their potential applications in quantum nanophotonics [15]. 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 [16–20]. 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 [21–33]. It is known that the nonlocal response causes a blueshift of the surface plasmon (SP) resonance of the metallic particle [26–29], and limits the SP field enhancement that in the LRA sometimes diverges [30–32]. We employ a simple generalization to the LRA, namely the hydrodynamic Drude model (HDM) [21, 22], 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 [16–20]. 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], where already the scattering potentials associated with the metal constituents are nonlocal. We recently showed that material nonlocal response plays an important role on HMMs in the limit of vanishing unit-cell size [33]. In particular, the nonlocal response gives rise to a cutoff to the hyperbolic dispersion curve, and an associated finite but very large fundamental upper limit to the enhanced local optical density of states (LDOS), which in the LRA is known to diverge in the limit of vanishingly small unit cells. Now in realized HMMs, it is predominantly the finite size of the unit cell that keeps the LDOS finite. So it is an intriguing question, not explored yet as far as we know, whether nonlocal response can also have noticeable effects in state-of-the-art HMMs. This theoretical paper describes our search for observable nonlocal effects in HMMs with unit cells much smaller than an optical wavelength, but much larger than the wavelength of hydrodynamic longitudinal pressure waves. 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 ε d . The metal is described in the HDM as a free-electron plasma with [21, 22, 26] ε T m(ω)=1 − ω 2 p ω 2 + iωγ , εL m(k,ω)=1 − ω 2 p ω 2 + iωγ − β 2 k 2 , (1) where ε T m is the Drude permittivity for the transverse electric fields as in the LRA, while the wavevector dependence of the permittivity ε L m of the longitudinal electric fields is responsible for the nonlocal response. We apply the hydrodynamic generalization of the common transfer-matrix method for lay- #189117 - $15.00 USD Received 18 Apr 2013; revised 14 May 2013; accepted 15 May 2013; published 17 Jun 2013 (C) 2013 OSA 17 June 2013 | Vol. 21, No. 12 | DOI:10.1364/OE.21.015026 | OPTICS EXPRESS 15028
Page 1: Hyperbolic metamaterial lens with h
Page 5 and 6: where ε loc ‖ and ε loc ⊥ are
Page 7 and 8: Figure 1(a) shows the value of k 0
Page 9 and 10: Fig. 4. Transmitted electric-field
Page 11: Local Nonlocal 0.465 0.465 -1 -0.5

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