Fig. 1a presents such a realization of the adiabatic lens in the form of a double-surface structure. A metallic sphere is placed in nanometer proximity of a second smooth metallic surface. From an electromagnetic point of view, the system morphs from a tapered metal-insulator-metal waveguide^15–17 in the gap region, to a single plasmonic interface in the upper hemisphere. As the surface plasmon waves originated from an object in the gap region propagate along the curved interface, from the gap to the upper hemisphere, they experience adiabatic decompression; the effective wavelength continuously increases from single nanometers in the gap region to the vacuum wavelength of hundreds of nanometers. The structure shown in Fig.1a supports both quasi-bound and radiation modes. A source placed in proximity of the metallic surface is capable of exciting two kinds of modes, naturally suggesting two different imaging approaches. One utilizes the interference of the plasmon polariton on the surface of the metallic sphere (Fig. 1); the other approach uses an elliptical reflector to interfere the decompressed EM wave to form an image on a far-field imaging plate (Fig. 2).

We emphasize that the super-resolution is achieved not through a sub-diffraction point-spread function (PSF)^{16, 17, 18}, which may require a perfect drain to be realized ¹⁹, but through a giant magnification derived from the decompression of EM waves. In an imaging system shown in Fig 1a, the wavelength is the fundamental unit that measures the distance between two points. The proposed double-surface system doesn’t have a fixed global wavelength but a spatial grid of variable wavelengths, depending on the distance between the two interfaces. The optical distance between two object points is preserved in image space, when measured in terms of the local wavelength. Because of the adiabatic decompression, that preserves the phase information, the physical distance between the two image points will be magnified as a consequence of the local wavelength dilation from the object to the image space (Fig.1a).

To elucidate the mechanism of the super-resolution imaging process, we developed an analytical model for a double-interface plasmonic system. It has been demonstrated that on a single sphere in isolation, surface waves generated by a point source can be refocused at the antipodal point forming an image of the source^{16, 17, 18}. This is a consequence of the constant positive intrinsic curvature of this geometry. However, the resolution in this case is limited by the wavelength of the surface wave, unless a perfect drain is placed at the imaging point¹⁹. If the metallic sphere is in contact at one point with a second metallic interface, in the absence of losses and nonlocal effects, the effective index would diverge at the point of contact⁵. The effective index is continuous and monotonic function of the distance between the two interfaces, and decreases as the surface mode is transformed from a gap-plasmon to a single-interface surface plasmon polariton (Fig. 1b). Such behavior can be described in terms of the following equation (derived in the Supplementary Material) for a disturbance propagating on a spherical surface of radius R ≫ λ and under the adiabatic condition ∂k_s/∂s ≪ k_s/λ: (0.1)d2Um(s)ds2+1Rtan(sR)dUm(s)ds+[ks2(s)-(mR)2sin2(sR)]Um(s)=0

An azimuthal dependence of the form e^imϕ is assumed. The surface coordinate s is simply related to the polar angle θ through s = Rθ. The function k_s(s) represents a coordinate dependent propagation vector tangent to the surface, which accounts for the variation of the local index as a consequence of the geometric potential induced by the interaction between the two spheres. A general solution to equation (0.1) is not readily available, nevertheless simple closed form solutions can be found for a few specific k_s(s) profiles to illustrate the concept of the adiabatic lens. In the case of a single sphere, the wave vector k_p approaches the value of a plasmon at a single interface, and equation (0.1) reduces to the well-known associated Legendre’s equation. In this case the solution is given in terms of associated Legendre’s functions of the first kind. More interesting is a profile of the form: (0.2)ks(s)={kpsin(s/R);0<s≤πR/2kp;πR/2<s≤πR

Notice that the specific profile (0.2) retains the properties of the local refractive index increasing monotonically from the common plasmon value k_p to a positive singularity at the contact at s = 0. The solution is then amenable to the following closed form: (0.3)Um(s)={am(s)cos[kp2R2-m2log(tans2R)]+bm(s)sin[kp2R2-m2log(tans2R)];0<s≤πR2cmP1+4kp2R2-12m[cos(sR)];πR2<s≤πR

Corrections to the field amplitude stemming from the wave decompression can be obtained by considering the radial dependence of the solutions and imposing power-flow conservation. Equation (0.1) can be cast in the canonical Sturm-Liouville form; from the knowledge of the set (0.3) it is possible to obtain the Green’s function of the system (see Supplementary Material), which yields the field distribution in the case of point source excitation shown in Fig. 1(c). A point source placed in the lower hemisphere at some angular distance θ₀ from the contact point at the South Pole (s=0), launches a surface wave that is refocused in the upper hemisphere at an angular position θ₁>θ₀ from the North Pole (s=π/2). This is a direct consequence of the wavelength dilation and effectively results in an angle-dependent magnification effect along θ.

This analytical solution elucidates an important aspect of the proposed imaging system: even though the smallest length scale of the system is far below the vacuum wavelength of the light, at every point in space, the system works above the local wavelength, meaning that a Hamiltonian optics formulation can be used to describe the system in terms conventional rays orthogonal to the phase fronts, in contradistinction with the sub-wavelength “Poynting vector rays” characteristic of hyperbolic metamaterials²³. In particular the index profile associated with (0.2) yields geodesics trajectories in the following analytical form: (0.4)ϕ(θ)=ϕ0+[ϕ(π2)-ϕ0]log(θ2)log(θ02)

The parameter ϕ₀ is the azimuthal coordinate at the source point located at elevation θ₀. The subset of the geodesics (0.4) contributing to the formation of an image is illustrated in the Supplementary Fig. 2, along with the ray envelope (shown in yellow) that encloses them.

Although the profile (0.2) captures qualitatively well the physics involved in the image formation in the type of systems described here, we would like to stress that the principles outlined are not related to a specific geometry, but are in fact applicable to a more general class of systems supporting guided modes and meeting an adiabatic condition for the variation of the local effective index. We performed full-wave numerical simulations to prove the proposed super-resolution imaging mechanism in the case of two spheres, which are not in contact in the supplementary materials.

The approach described so far relies on the guided surface modes excited by a source located in the gap between two metallic interfaces. A complementary mode of operation exploits the scattered angular spectrum instead, and can be integrated with conventional imaging systems. Exploiting an elliptical reflector, here we show in fact that a double-sphere plasmonic system can enhance the resolution of a conventional optics system directly (Fig. 2). One focus of the ellipsoid reflector coincides with the center of the gap between the two spheres. An imaging plane is placed at the other focus of the elliptical reflector. A point dipole is placed within the gap and emits light in all directions. Our simulation shows that, along each direction, the light experiences adiabatic decompression. The elliptical reflector collects the lights, and interfere them at the image plate. If the light source is at the center of the gap, all of the light paths will accumulate equal phase at the center of the image plane. The constructive interference gives rise to a point image (Fig. 2b). If we move the light source away from the center, light rays along different directions accumulate different phases. The point image will move away from the center point accordingly (Figs. 2c and d). A displacement of a point source in the gap as small as a few nanometers is clearly resolvable at the image plate, which demonstrates the super-resolution of the system. The resolution depends on the shortest local wavelength that the system can achieve in the gap; therefore decreasing the gap size increases the resolution (Figs. 2i–k).

In Figs. 3a and b we show that two coherent point sources located 48 nm apart can clearly be resolved at the image plate. As the gap increases to 24 nm, the two points are no longer resolvable (Fig. 3c). The resolution limit of this type of system depends on the extent to which one can slow down the light, and it is ultimately determined by the loss of the material. As shown in Fig. 3e, the loss reduces the magnification as well as increases the width of the image peak. Therefore, we estimate that the practical limit of the resolution that the adiabatic lens can achieve will be limited to the skin-depth of the material that is used, which is about tens of nanometers in visible range. A possible method to further circumvent this limit could be by using active materials that uses gain media to compensate the loss^{20, 21, 22, 23, 24}.