Broadband superoscillation brings a wave into perfect three-dimensional focus

The fundamental properties of a wave precludes it from being localized to subwavelength distances in all dimensions of the wave's existence. The inability to focus electromagnetic waves to an all-direction subwavelength spot limits the 3D resolution of a conventional imaging system to about half the imaging wavelength. A plethora of super-resolution imaging systems have been designed which obtain super-resolution in one or two (but not all) dimensions, but they suffer various restrictions in working distance and the classes of objects they can image. In this paper, we report a first investigation into a wave that is focused to subwavelength dimensions in all directions. After reviewing the physics of wave dispersion and diffraction which seemingly preclude this phenomenon, we sidestep these preclusions using a broadband superoscillation waveform and synthesize an all-direction subwavelength focus. We report the salient spatial and temporal features of this wave, and apply it to achieve 3D super-resolution imaging.

Figure 1

3D subwavelength focusing for a broadband waveform. (a) A diagram of spectral components involved. Each frequency component forms a spherical shell in $k$ space. (b) Weights $A_n$ for spherical Bessel functions (zeroth order, first kind) which would superimpose to form a 3D subwavelength focus. (c) The resultant electric field amplitude ($x$-axis cut). The waveform is radially symmetric. (d) A closeup of the superoscillation region (blue, solid) plotted alongside the fastest varying spherical Bessel function $j_0\left(k_{m}r\right)$ (red, dashes). Subwavelength focusing to 66% of the diffraction limit is observed.

Figure 2

Temporal waveform pulsation. (a) The time evolution of the waveform amplitude along the $x$ axis. The waveform is normalized to show constant strength at the principal peak at $x=0$. The blue dotted line denotes the 3 dB width of the 3D focal spot. (b) Waveform amplitude profiles at four different times: $t=t_0=0$ (blue, solid), $t=0.005T_B$ (black, dash), $t=0.01T_B$ (red, dash-dot), and $t=0.2T_B$ (green, dotted). The spot width at $t=0.2T_B$ is indicative of the waveform width for most of the period, when superoscillatory wave features are out of focus.

Figure 3

Super-resolution imaging using an all-direction subwavelength focus. (a) A schematic showing the geometry of four point objects in the numerical experiment. The positions of the objects are $\mathrm{A}(0,-s/2\sqrt{6},s/\sqrt{3})$, $\mathrm{B}(-s/2,-s/2\sqrt{6},-s/2\sqrt{3})$, $\mathrm{C}(s/2,-s/2\sqrt{6},-s/2\sqrt{3})$, and $\mathrm{D}(0,-3s/2\sqrt{6},0)$. (b) The superoscillation waveform used in the experiment, plotted with the spherical Bessel function $b_0\left(kr\right)$. Clearly, the superoscillation waveform contains an all-direction subwavelength focus and lower sidelobe levels.

Figure 4

Super-resolution imaging using an all-direction subwavelength focus. (a) Diffraction-limited images obtained for the cluster shown in Fig. 3, along the plane $y=-s/2\sqrt{6}$; (b) super-resolution images equivalent of (a). (c) Diffraction-limited images obtained for the cluster shown in Fig. 3, along the plane $x=0$; (d) super-resolution images equivalent of (c). The separation distances are $s_1={\lambda }_m$ (top row), $s_2=0.7{\lambda }_m$ (middle row), and $s_3=0.4{\lambda }_m$ (bottom row), where ${\lambda }_m=2\pi /k_m$ represents the shortest wavelength involved in the imaging process.

Figure 5

Sensitivity analysis for superoscillation-based focusing and imaging. (a) Typical waveform variations (black, dashed) when a random-phased white Gaussian noise is added to the sub-wavelength focus shown in Fig. 1. Variations are shown for $\text{SNR}=15$ dB. The original waveform is also shown for comparison (blue, solid). (b) $y$-plane (top-row) and $x$-plane (bottom-row) super-resolution images obtained in the presence of noise, with $\text{SNR}=10$ dB (left), 5 dB (middle), and 2 dB (right). For this case the source separation is $s=0.7{\lambda }_m$. (c) $y$-plane (top-row) and $x$-plane (bottom-row) diffraction-limited images in the presence of noise, with $\text{SNR}=10$ dB (left), 5 dB (middle), and 2 dB (right). The source separation is the same as in (b).

Figure 6

Comparison between 3D and 2D super-resolution imaging. This figure shows imaging results for three different scenarios of single source (top row), three source on the same plane (middle row), and four sources as shown in Fig. 3. For the second and third cases the separation is $s=0.7{\lambda }_m$; all images are taken along a constant-$y$ plane shown on the left panel. The middle column displays images processed by a 2D superoscillation filter; the right column displays images processed by a 3D superoscillation filter.