Investigating the equivalent numerical aperture of super-oscillation metalenses

Optical microscopy bridges the human world and the microscopic realm, being widely used in both biological and non-biological fields. Optical super-oscillation, a phenomenon where interference among a series of low-frequency light waves produces a rapidly oscillating field in a localized far-field region, enables breaking the diffraction limit. It has been demonstrated that far-field super-resolution focusing can be realized with a super-oscillatory metalens. While the numerical aperture (NA) of a conventional refractive lens directly reflects the transverse spatial cutoff frequency of the focused optical field as f_c = NA/λ, the super-resolution metalens is a diffractive device whose generated optical field cutoff frequency does not simply depend on the lens NA. This paper presents a study on the spatial frequency spectrum of the focused optical field of metalenses. Four metalenses, all with an NA of 0.93 but differing in design method and cutoff frequency, are designed and fabricated. Both simulations and experiments validate that our proposed equivalent NAs (NA^(f)_equ and NA^(s)_equ) can effectively reflect the highest effective frequency of the lens-focused optical field and focal spot size, offering a promising tool for analyzing the focusing performance of super-oscillatory lenses.