Permissions for Reuse

The maximum achievable resolution of any conventional lens, the function of which is based on focusing of electromagnetic radiation, is of the order of the wavelength of the used radiation (the limit that is also known as the diffraction limit or the Rayleigh limit). This limit directly arises from the low-pass spatial bandwidth constraints of conventional optical systems.¹ Indeed, for an objective immersed in a medium of refractive index n the numerical aperture is NA=n sin _max and the spectrum of transversal wave numbers of the incident field the objective reacts to is limited by 0|k_t|2 NA/. The evanescent waves with |k_t|>2n/ that carry information about the subwavelength details of the source are lost in the conventional optical systems.

The resolution of an imaging device can be greatly improved if this device is able to react to the evanescent part of the spatial spectrum of the incident field. There are several known techniques that allow subwavelength-resolution imaging: superlenses based on the use of materials with negative parameters,^2,3 superlenses based on phase-conjugation (nonlinear three-wave mixing),⁴ and arrays of small resonant particles.^5,6,7,8,9 However, the only means to actually measure the image-plane near-field distributions at the subwavelength scale is the scanning near-field microscope that uses a small moving probe. Recently, hyperlenses, which transform evanescent modes into propagating ones, have been proposed^10,11,12 with the goal to develop subwavelength microscopes. These devices allow image detection by a stationary system like a charge-coupled device matrix, however, the manufacturing technology for hyperlens structures needs further development.

In this paper we introduce a near-field subwavelength imaging device that is based on frequency scanning with following postprocessing of measured data. The field is measured only at one or two points in space and the role of sensor is played by an electrically dense grid of small resonant particles. The device does not use any moving probes, allowing very fast subwavelength imaging. In this device the conventional measuring of near fields at many points in space is replaced by measuring resonantly enhanced near fields at many frequency points, which allows us to calculate the spatial field distribution. The maximum spatial resolution of the device is limited by the period of the sensing grid and depends on the quality factor of grid particles as well on the number of particles.

The present method is a development of the approach of our previous paper,⁵ which introduced a system of two coupled resonant grids or arrays of small inclusions as a superlens capable of resonant amplification of evanescent fields and creation of images with subwavelength detail. That idea was further extended to enlarging superlenses.⁶ Possible realization in the optical region was also demonstrated.^7,8 Experimental confirmations in the microwave region were made using grids of small resonant electric dipoles^5,6 and magnetic dipoles (split rings).⁹ Grids of resonant particles get strongly excited when the transversal wave number k_t of an incident evanescent wave matches the propagation factor of the grid's surface wave. We showed both theoretically and experimentally how this effect can be used for resonant amplification of evanescent fields. Because of the resonance, the amplitude of the evanescent wave in the image plane of a dual-grid structure⁵ becomes equal to the amplitude of the evanescent incident wave in the source plane.

Matching of the propagation factors takes place at a certain frequency. The grid of densely packed resonant particles can be modeled by its grid impedance Z_g(,k_t), which connects the surface-averaged tangential electric and magnetic fields.¹³ The grid impedance depends on the frequency and the propagation factor along the grid plane k_t. The dispersion equation for the surface waves on an impedance grid in free space can be written as

where Z₀(,k_t) is the wave impedance of the corresponding free-space plane wave

where ₀= and k₀=. The solutions of this equation are pairs (,k_t) that lie on the grid's dispersion curve. If the grid is periodic with the period d₀ (₀ is the free-space wavelength), its dispersion curve usually looks like what is shown in Fig. 1(a).

Figure 1.

We see that for an operating frequency [_min,_max] there is only a single resonating value for the propagation constant k_t. By scanning the frequency from _min up to _max, we can sequentially excite the surface modes of all possible k_t ranging from 0 to k_{t max}=/d. The amplitude of an excited mode is proportional to the amplitude of the corresponding spectral component of the incident field. Knowing the spatial profile of every surface mode (from the theory or from an initial calibration measurement) we can find the amplitude of the mode from a field measurement done only at a single point in the image plane (or at most at a couple of discrete points, to account for degenerate cases of zero field of certain modes at the measurement point). In a finite-size grid the spectrum of surface modes is discrete. In this case the field measurement is done at discrete frequencies.

It is worth noting that for open structures (in contrast with setups confined inside a closed impenetrable cavity) the surface modes with 0k_tk₀/d are leaky modes, i.e., they radiate into surrounding space. However, the operation of the microscope is based on the use of eigenmodes with high propagation constants (k_tk₀), which are strongly bound to the grid surface and radiate only at inhomogeneities and at the grid ends. Moreover, in some resonant systems, such as the metasolenoid,¹⁴ even the modes with small values of k_t produce a strong resonant response because of a high Q-factor of its resonances (Q is of the order of 10³).

In a system of two coupled resonant grids⁵ the amplitude of the field in the image plane can be made the same as in the source plane so that the system works as a “perfect lens”² at every resonant frequency of the system. This allows us to find the spatial field distribution of the unknown source by weighted integration of measured values, as is explained in Sec. II. In principle one can use just a single resonating grid. The only difficulty here is that the relation between the amplitude of the excited surface mode and the unknown external field is more complex. The single-grid microscope is explained in Sec. III.