Luo, Johnson, Joannopoulos, and Pendry [Phys. Rev. B 65, 201104(R) (2002)] showed that a photonic crystal whose band has a convex, frequency-shrinking equal-frequency contour refracts light negatively at all incident angles — all-angle negative refraction (AANR) — without the bulk medium possessing a negative effective (homogenized) refractive index, and they designed and numerically demonstrated a flat 'microsuperlens' based on this mechanism at microwave frequencies. They conjectured that the effect generalizes to fully three-dimensionally periodic crystals and to optical frequencies, where the metallic components of the microwave demonstrations cannot be reused, while acknowledging that true single-beam refraction in fully 3D systems is difficult and that the imaging limitations (surface reflectivity, slab termination, finite thickness, aberrations), achievable AANR bandwidth, and tolerance to fabrication imperfection were unquantified. This problem asks for that conjecture to be established: single-beam AANR without a negative effective index in a fully three-dimensional photonic crystal at optical frequencies, together with a superlens implementation whose imaging performance, bandwidth, and robustness to disorder are quantified. No reported work demonstrates the stated combination: adjacent results — 3D microwave flat lenses that use a negative effective index, two-dimensional near-infrared negative refraction and imaging, mid-infrared 3D dispersion mapping, broadband and reflectionless AANR designs, and mechanism-distinct hyperbolic, polaritonic, and topological routes — leave it open, and so does the numerically proposed three-dimensional optical AANR crystal of Ao and He (reference 4), which rests on a design and a focusing-imaging simulation rather than a realized, characterized superlens.

Background

In a conventional dielectric, a refracted beam crosses to the opposite side of the surface normal. Veselago media, with simultaneously negative electric permittivity and magnetic permeability, instead have a negative refractive index and refract beams to the same side of the normal; Pendry showed that such a negative-index slab can act as a 'perfect lens' that refocuses evanescent waves and beats the diffraction limit. Photonic crystals offer a different route: they are dielectrics periodic on the scale of the wavelength of light, whose allowed modes form photonic bands. Each band has a dispersion surface, and its intersection with a plane of constant frequency is the equal-frequency contour (EFC). Refraction at an interface is governed by conservation of the wave-vector component parallel to the surface together with the EFC shape, and the energy (group) velocity is everywhere perpendicular to the EFC. If an EFC is convex and shrinks as frequency increases, the group velocity refracts negatively for every incident angle within the accessible numerical aperture — the all-angle negative refraction (AANR) criterion of Luo, Johnson, Joannopoulos, and Pendry. Crucially, this needs no negative effective index: the band in question can belong to a medium whose long-wavelength (homogenized) refractive index is positive, so the effect does not inherit the strong dissipation and narrow bandwidth of resonant negative-index metamaterials. In the 2002 paper the authors gave simple geometric criteria for AANR, designed a flat 'microsuperlens' exploiting them, and demonstrated it numerically; the same group's companion experiment realized the mechanism at microwave frequencies with a two-dimensional array of metallic elements, and metallic photonic crystals were further analyzed by them in 2003. The paper left its own generalizations open: whether AANR without a negative effective index survives in fully three-dimensionally periodic crystals — where the periodic lattice can couple an incoming wave into several propagating diffraction orders (each a Bloch mode of the crystal carrying the same conserved parallel wave-vector component), so achieving true single-beam refraction is difficult — whether it can be scaled to optical frequencies, where the microwave metallic structures are not available and dielectric multilayer films or index-guiding (confining light in a slab waveguide whose in-plane periodicity supplies the dispersion engineering) were the suggested substitutes, and what imaging quality, bandwidth, and fabrication tolerance a practical optical superlens built this way would have. Several neighboring routes have since been developed and must be distinguished from this mechanism: hyperbolic (indefinite) media, whose anisotropic permittivity tensor has components of mixed sign and which also yield all-angle negative refraction; flat lensing by self-collimation, in which a flat band collimates beams without any refraction of the negative type; and topological Weyl-metamaterial proposals that suppress interface reflection by topological protection. Negative refraction has also been observed directly in two-dimensional photonic-crystal slabs at near-infrared (telecommunication) wavelengths. None of these is the positive-effective-index, convex-EFC mechanism in a fully 3D periodic dielectric at optical frequencies.

Problem Statement

Can the all-angle negative refraction (AANR) mechanism of Luo, Johnson, Joannopoulos, and Pendry — negative refraction for every incident angle produced by a convex, frequency-shrinking equal-frequency contour (the constant-frequency cross-section of a photonic band's dispersion surface) of a photonic band, with a positive bulk effective index and no resonant negative-index components — be carried from its two-dimensional microwave realization to fully three-dimensional periodic structures and to optical frequencies? Specifically: (i) exhibit a fully three-dimensionally periodic, fabricable photonic crystal whose band structure yields AANR through this positive-index convex-contour mechanism, with true single-beam negative refraction at its surfaces (suppression of competing Bloch diffraction orders and control of surface termination); (ii) realize this behavior at optical frequencies (visible or near-infrared), where the metallic elements of the microwave demonstrations cannot be scaled; and (iii) on such a realization, build and characterize a flat superlens: determine its imaging performance including surface reflectivity, finite-thickness and slab-termination effects, and aberrations, and quantify the achievable AANR bandwidth and the tolerance to fabrication imperfections such as structural disorder and finite sample size. Sub-goals (i)-(iii) share one objective: establishing that the without-negative-index AANR mechanism is a viable route to a practical optical superlens.

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