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Negative refraction and sub-wavelength focusing in the visible range using transparent metallo-dielectric stacks

Michael Scalora, Giuseppe D’Aguanno, Nadia Mattiucci, Mark J. Bloemer, Domenico de Ceglia, Marco Centini, Antonio Mandatori, Concita Sibilia, Neset Akozbek, Mirko G. Cappeddu, Mark Fowler, and Joseph W. Haus

Open Access Open Access

Abstract

We numerically demonstrate negative refraction of the Poynting vector and sub-wavelength focusing in the visible part of the spectrum using a transparent multilayer, metallo-dielectric photonic band gap structure. Our results reveal that in the wavelength regime of interest evanescent waves are not transmitted by the structure, and that the main underlying physical mechanisms for sub-wavelength focusing are resonance tunneling, field localization, and propagation effects. These structures offer several advantages: tunability and high transmittance (50% or better) across the visible and near IR ranges; large object-image distances, with image planes located beyond the range where the evanescent waves have decayed. From a practical point of view, our findings point to a simpler way to fabricate a material that exhibits negative refraction and maintains high transparency across a broad wavelength range. Transparent metallo-dielectric stacks also provide an opportunity to expand the exploration of wave propagation phenomena in metals, both in the linear and nonlinear regimes.

©2007 Optical Society of America

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Figures (13)
Fig. 1.
Fig. 1. Transmittance vs. separation distance between two metal layers 32nm thick, for a dielectric spacer medium having n=2 (inset), and incident wavelength of 500nm. The resonance tunneling condition occurs for dielectric layer thickness of 72nm, or 0.29λ, where λ is the wavelength in the material.

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Fig. 2.
Fig. 2. Plane-wave transmittance vs. wavelength at normal incidence from a symmetric, 13-layer stack composed of Ag(32nm)/X(21nm), inclusive of entry and exit X layers 11nm thick (transparent metal), and from a periodic stack composed of 6 periods of Ag(32nm)/X(21nm). Halving the thickness of first and last layers increases transmittance significantly across the transparency range, and affects field localization properties (Fig. 8 below).

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Fig. 3.
Fig. 3. A Gaussian, TM-polarized wave packet is incident at 45° on the transparent metal stack described in Fig. 2. The figure shows several snapshots of the magnetic field intensity. The centroid of the pulse that exits to the right of the stack is shifted upward by approximately 266nm. Plane-wave reflectance is ∼5% at 400nm.

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Fig. 4.
Fig. 4. A TM-polarized beam or pulse is incident from vacuum (or other medium with positive permittivity) on a metal layer, at frequencies below the plasma frequency where the real part of its dielectric constant is negative. The x inside the circle indicates that the H field points inside the page. Then, preservation of the continuity of the longitudinal component of the displacement field, D out z = D in z , requires that ε out E out z = ε in E in z . As a result, a sign change of the field Ez occurs when the dielectric constants have opposite signs.

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Fig. 5.
Fig. 5. Schematic representation of the refraction that occurs inside the stack. P T is the total, averaged momentum inside the stack. Upper right: the local momenta inside two adjacent metal and dielectric layers are shown. The local momentum density, i.e. the Poynting vector, generally differs from the total momentum within a given layer. The refraction process should be viewed from a global perspective, by collecting information across the entire layer.

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Fig. 6.
Fig. 6. Schematic representation of twin momentum vectors that lead to the formation of internal and external foci, based on the results depicted in Fig. 5. In this picture the location of both foci are approximate, as the averaging process neglects effects of field curvature. Both internal and external focal points generally depend on the slit-stack distance.

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Fig. 7.
Fig. 7. Negative refraction angle as a function of incident wavelength. The incident angle is fixed at 45°. The angle decreases as the carrier wavelength is increased. This is due in large part to the metal dispersion, which causes a drop in the magnetic field intensity inside the metal layers only, resulting in a reduction of anomalous momentum.

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Fig. 8.
Fig. 8. On-axis ∣E∣2 and ∣H∣2 vs. position inside the chirped stack described in Fig. 2, for a field incident from the left. The real part of the dielectric constant alternates between the values of 16 and -3.77 (thin, black curve; right axis). The fields are unusually intense inside each metal layer, leading to large energy and momentum values inside each metal layer. At 500nm, for silver Re(ε)= -8.57. The shape and amplitude of the electric field intensity change little across the stack. While the shape of ∣H∣2 remains almost identical, it decreases by an average factor of 3 only inside the metal layers, causing a drop in stored anomalous momentum, and a consequent reduction of the negative refraction angle.

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Fig. 9.
Fig. 9. A quasi-monochromatic Gaussian wave packet is incident from the bottom on a 140nm-thick germanium substrate with an aperture ∼125nm wide. The transparent metal stack is described in the caption of Fig. 2, and is located ∼50nm away from the slit. The distance to the collection point (screen) may vary.

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Fig. 10.
Fig. 10. Bird’s eye-view of a snapshot of the magnetic field intensity inside and passed the stack. A focal point is clearly visible outside the stack.

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Fig. 11.
Fig. 11. Image produced by the slit on the image plane indicated on Fig. 9. The full width at half maximum of the H-field that propagates through the stack is ∼200nm and it is roughly five times narrower compared to the same field propagating in free space.

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Fig. 12.
Fig. 12. Bird’s eye-view of a snapshot of the magnetic field intensity inside and passed the transparent metal stack. The image is produced by two 125nm apertures located on the Ge substrate, having a center-to-center distance of ∼325nm. The slits are resolved with a visibility of approximately 40%, yielding diffraction-limited, sub-wavelength resolution. A third bright spot, Poisson’ spot, appears down-range, as a result of constructive interference between the primary spots, which in turn become secondary sources in the Poisson’ spot formation process.

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Fig. 13.
Fig. 13. Sz(ky/k0,z) is the Fourier transform of the longitudinal Poynting vector Sz(y,z). The contiguous arrows to the left indicate the beginning and the end of the stack, and where vacuum begins. This spectral snapshot reveals the longitudinal dynamics of each transverse k-vector. It is evident that in this example evanescent wave vectors (ky/k0>1) are hardly excited, an indication that in this regime surface waves are not supported.

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Equations (1)

Equations on this page are rendered with MathJax. Learn more.

P ( t ) = ∫ z = − ∞ z = ∞ ∫ y = − ∞ y = ∞ S y z t c 2 dy dz = ∫ z = − ∞ z = ∞ ∫ y = − ∞ y = ∞ E x H 4 πc dy dz
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