Sub-Diffraction Focusing Using Metamaterial-Based Terahertz Super-Oscillatory Lens

Abstract

This paper presents a metamaterial-based super-oscillatory lens (SOL) fabricated by photolithography on a glass substrate and designed to operate at sub-terahertz (sub-THz) frequencies. The lens consists of repeating crisscross patterns of five-ring slits with sub-wavelength diameter. The lens is capable of generating multiple focal points smaller than the diffraction limit, thereby allowing many points to be inspected simultaneously with sub-wavelength resolution. After elucidating the influence of the lens parameters on light collection through calculations by the finite element method, the fabricated lens was then evaluated through actual experiments and found to have a focal length of 7.5 mm (2.5λ) and a hot spot size of 2.01 mm (0.67λ) at 0.1 THz (λ = 3 mm), which is 0.27 times the diffraction limit of the lens. This demonstrated sub-diffraction focusing capability is highly effective for industrial inspection applications utilizing terahertz waves.

1. Introduction

In recent years, the number of applications using terahertz (THz) and millimeter waves has continued to increase, particularly for 6 G technology in telecommunications [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]. The THz frequency band (from 0.1 to 10 THz) is also expected to be highly effective for industrial product inspections, as THz waves have the potential to detect cracks in asphalt walls and foreign metallic objects in products. However, THz applications are currently limited by resolution constraints.

Since the THz wavelengths (from 30 μm to 3000 μm) are longer than the visible and near-infrared light wavelengths (from 0.360 μm to 1.4 μm) conventionally used for inspection, the resolution of THz-based inspection cannot be as small as that of visible light when using ordinary lenses. This is known as the diffraction limit. Terahertz lenses are generally made of polytetrafluoroethylene (PTFE), polyethylene, silicon, etc., and the spot size made by these lenses is restricted by the diffraction limit δ:

$$\delta =0.61\lambda /\mathrm{NA}$$

(1)

where λ is the wavelength in THz. NA is the numerical aperture of the lens, which is determined by

$$\mathrm{NA} =n \sin \theta$$

(2)

where n is the refractive index of air (the medium surrounding the lens) and θ is the maximum half-angle of the light cone defined by an apex at the focal point and a base on the lens.

Because of the evolving strict inspection resolution requirements, methods to achieve higher resolution are required. Various methods have been studied to achieve subwavelength resolution in the THz frequency band [16,17,18,19,20], including superlenses and bull’s-eye lenses [17,18]. These methods use evanescent waves to achieve resolution at the expense of the working distance (WD). Though trivial for microscopy applications, a short WD can limit the applicability of industrial product inspection methods. A few metamaterial-based lenses with negative refractive indices and which also utilize evanescent waves have also been reported [19,20]. In this work, we focused on other metamaterial approaches to achieve subwavelength resolution and a long WD. Some studies have revealed that the diffraction of light using a binary mask produces a subwavelength spot that allows imaging with a higher resolution than that of conventional lenses [21,22,23]. This physical phenomenon is known as super-oscillation, in which band-limited functions can oscillate arbitrarily faster than the highest contained Fourier component. In 2006, Berry and Popescu were the first to propose that light fields diffused through subwavelength gratings can generate arbitrarily small spatial energy regions without resorting to evanescent waves, theoretically proving the feasibility of resolution enhancement in imaging systems using super-oscillation methods [24]. Recently, Zheludev demonstrated this phenomenon for the first time by observing a focal spot below the diffraction limit generated by a quasiperiodic metallic nanohole array mask [25]. Since then, various devices and design methods for generating super-oscillating electric fields have been proposed one after another. A lens that employs this phenomenon was developed to achieve subwavelength resolution in the visible and ultraviolet (UV) light regions, called “Super-oscillatory lenses (SOLs)” [26,27].

One technique in SOL design is to control the interference of incident waves by optimizing the pattern of concentric ring slits. However, this method requires complex design calculations [28]. Moreover, the lens produces only one focal point at the focal length. On the other hand, a binary mask with multiple holes smaller than the wavelength [29] reportedly achieves a similar effect by controlling the transmission and retardation of the incident wave using metamaterials [30], which is simpler to design than the former. Roy et al. [30] arranged ring slits in a crisscross pattern for visible light, which generated multiple tiny spots when the electromagnetic waves passed through the surrounding slits faster than those passing through the central slit. In metamaterial-based SOLs, WD is unbounded by the lens vicinity because no negative refractive index is used. Moreover, this lens produces a focal point directly above each unit pattern (Figure 1). This particular characteristic is attractive for industrial product inspection as it offers the advantage of allowing multiple points to be inspected at the same time. In contrast, conventional lenses only focus on a single point and consequently require longer times to inspect the entire surface of a wide product, such as a film. Hence, a multiple-focal-points-generating lens can significantly shorten the inspection time.

In this study, we fabricated and evaluated a metamaterial-based sub-THz-band SOL using a unit cross pattern smaller than 3 mm, which is the wavelength that corresponds to 0.1 THz, for the first time. Moreover, it was experimentally verified that such a lens is useful for inspection using THz in industrial applications. We demonstrated that a spot with a diameter of 0.67λ, which is 0.27 times the diffraction limit, could be generated by the SOL at a position farther from the near field.

2. Materials and Methods

2.1. Design and Fabrication

Metamaterial is a technology that uses periodic patterns smaller than electromagnetic wavelengths to achieve unique outcomes. When a plane wave is normally incident on a lens with evenly arranged circular slits, the transmitted wave pattern causes no phase delay. However, when five circular slits are arranged in a crisscross pattern on the lens, as shown in Figure 2a, a phase delay occurs, and the waves are focused at focal length from this pattern. This can be explained by the diffraction effect, otherwise called the Talbot effect [31]. The grating image is repeated at regular distances away from the grating plane when the plane wave is incident on the periodic diffraction grating. The distance at which a spot is generated can be expressed using the Talbot distance Z_T as follows:

$$Z_T=\lambda /\left(1-\surd \left(1-{\lambda }^2/a^2\right)\right)$$

(3)

where a is the diffraction grating period. The focal length can be adjusted by varying the wavelength and unit slit spacing. We designed two pattern types, circular and square, for subwavelength focusing with 0.1 THz (λ = 3 mm).

Figure 2 shows the schematic of the lens comprising circular and square unit patterns (Figure 2a,b, respectively). The yellow sections are nontransparent to THz waves. For an effective metamaterial, the unit slit must be sufficiently small relative to the wavelength. Thus, we designed the parameter D, which is the diameter of the circular patterns or the diagonal of the square patterns, as 690 μm (approximately 0.23λ), which is sufficiently small compared to 3 mm. The width of these patterns is 127.5 μm (approximately 0.04λ) and the distance l between each unit slit is 1275 μm. The distance A between the centers of each unit pattern was 3875 μm. With these dimensions, the focal length of the lens is 7.9 mm by design, as determined using Equation (3).

The performance of the aforementioned SOL was elucidated through calculations according to the configuration illustrated in Figure 1 and by considering a circularly polarized ideal plane wave as the incident THz beam. The results of the simulations are shown in Figure 3. The intensity distribution profiles using square and circular slits (Figure 3a,b, respectively) exhibit spots generated at the same focal length of approximately 8 mm. In addition, the focal spots are repeatedly generated along the direction perpendicular to the lens and the distance between spots is consistent with the calculated Talbot distance of 7.9 mm. The bottom graph of Figure 3b shows the phase distribution through the circular patterns.

The spot size as a function of the distance from the lens is plotted in Figure 3c, where blue and red dots represent the results for the square and circular patterns, respectively. The spot sizes generated by the two patterns are in good agreement although the circular patterns generated spots that are approximately three times more intense than those generated by the square patterns.

Figure 3d shows the intensity distribution due to the 3 × 3 circular unit patterns at a distance of 8 mm from the lens. The advantage of this type of metamaterial lens over conventional lenses can be clearly observed in these simulated results. While conventional lenses form only one spot at the focal length, each unit pattern of the SOL forms a spot at a focal distance. With this metamaterial lens, it is possible to access multiple inspection points simultaneously even though the intensity at each focal spot is just ~19.5% of the incident beam intensity as the energy is distributed over all the unit patterns on the lens.

In the case of using square unit patterns, the focusing is affected by the direction of the incident electromagnetic wave polarization. Figure 4 shows the calculated intensity distribution of linearly polarized light (along the y axis) incident on a square pattern lens, and this distribution is normalized by the maximum intensity of the focal spot. Figure 4a shows the intensity distribution on the plane parallel to the direction of polarization (yz plane), and Figure 4b shows the intensity distribution on the plane perpendicular to the direction of polarization (xz plane). These graphs show that the light is strongly focused on the plane perpendicular to the direction of polarization, generating an elliptical spot at a focal length of 7.9 mm. Figure 4c is a graph comparing the spot size where the blue and red lines show the sizes of the spot on the planes parallel and perpendicular to the direction of polarization, respectively. It can be clearly observed that the spot size on the plane perpendicular to the polarization is about half of the spot size in the parallel direction. This difference in resolution makes the square unit pattern difficult to use for product inspection. Thus, the circular unit pattern is preferred.

As already mentioned above, there are three important parameters for lens design: the width of the slits, the diameter of the ring D, and the distance between rings l. Figure 5 shows the results of calculations of how these three parameters affect the focusing ability. Figure 5a,b compare the light-gathering ability of three patterns that differ only in ring diameter. Figure 5a shows the intensity distribution after light passed through each lens. The upper graph shows the results using a lens with a ring diameter 20 μm smaller than the design value of 690 μm while the lower graph shows the results using a lens pattern with a ring diameter of 710 μm, which is 20 μm larger than the design value. As in Figure 3b, a spot appears around 7.9 mm in both results, and the focus repeatedly appears in the direction directly above the lens. Figure 5b compares the spot sizes generated by these lenses. The blue plot represents the spot sizes using a ring diameter of 670 μm, and the red plot shows the results for a ring diameter of 710 μm. The spot has a size of 0.63λ at a designed focal length of around 7.9 mm. Comparing Figure 3b with Figure 5a,b, it can be confirmed that the focal length at which the spot appears is the same as that of the designed lens because the lattice constant does not change when the ring diameter is changed, as determined from Equation (3).

Figure 5c shows the intensity distribution after light passes through each lens along x = 0 mm. The depth of field (DOF) [32], which is generally defined as the range of 80% of the spot’s central intensity, is 2.1 mm for the first spot and 2.45 mm for the second spot using either 670 μm- or 710 μm-diameter lenses, which are consistent with the results using the 690 μm-diameter lens. This confirms that the diameter of the ring does not affect the DOF, which is the range of light that can be focused by this SOL.

Next, we elucidated the focusing ability by changing only the slit width. Figure 5d,e show the results of calculations using lenses with slit widths 10 μm larger and 10 μm smaller than the designed value. The upper graph shows the results using a lens with a slit width of 117.5 μm, and the lower graph shows the results using a lens with a slit width of 137.5 μm. Figure 5e shows the spot size relative to the direction away from the lens. As in Figure 5a,b, the spot intensity varies with the slit width but the location and size of the spot are almost the same, indicating that the slit width has little effect on the focal spot size. The depth of focus is also similar, with little effect of slit width. However, unlike the change in ring diameter, the focused intensity weakens as the slit width increases. This is expected because the diffraction caused by the slit is weakened as the slit width increases.

Finally, the focusing ability of the lenses was compared by changing only the distance l between the rings, which is a lattice constant. Figure 5f shows the intensity distribution after light passes through the lenses when the distance l between the rings is either 20 μm shorter (upper graph) or 20 μm longer (lower graph) than the designed value. Figure 5g shows the spot size in the direction away from the lens, with the blue plot representing the results when the distance between the rings is shorter than the designed value and the red plot depicting the results when the distance between the rings is longer than the designed value. It can be seen that changing the distance between the rings does not affect the spot size. However, the positions where the spots are generated change depending on the spacing between the rings. The simulation results presented in Figure 5f,g are consistent with the Talbot distance formula, which shows that the focal length increases as the distance between the rings increases. The focal lengths obtained according to the Talbot distance formula are 7.6 mm when the distance l between the rings is 20 μm shorter than the designed value and 8.2 mm when l is 20 μm longer than the designed value.

Since the circular patterns exhibited significantly better focusing efficiency than the square ones, we fabricated the lens with circular unit patterns. The fabricated super-oscillatory lens, which has a diameter of 101.6 mm, is shown in Figure 6a and the schematic of its cross-section is depicted in Figure 6b. Figure 6c shows a microscopical image of the fabricated circular slit. The circular patterns were defined by photolithography on the surface of a 400 μm thick glass substrate by depositing 10 nm thick chromium and 300 nm thick gold. Since gold does not adhere well to oxides such as glass, chromium was deposited as a buffer layer between the glass substrate and the gold layer. The thickness of the gold layer was intentionally set thicker than the skin depth for the opaque parts of the lens to sufficiently block THz waves.

2.2. Experimental Setup

The schematic of the experimental setup is shown in Figure 7. A Keysight N5247A network analyzer was attached to a pair of N5256X10 WR-10 mm-wave extenders. This network analyzer for the 0.067 to 0.110 THz frequency range served as an emitter on one side and a detector on the other side. The dynamic range of this network analyzer is 88.5 dB. A circularly polarized incident beam was formed by a linear-to-circular polarizer or wave plate attached to the emitter extender, followed by a corrugated horn antenna to minimize the side lobe levels. A pair of conventional Teflon lenses, each with 150 mm diameter and 100 mm focal length, were used to collimate the antenna output onto the lens. A probe antenna with a 2.1 mm diameter aperture was attached to the detector extender through a second wave plate to capture the beam transmitted through the lens. To avoid the detection of reflected waves from the SOL, an absorber was placed at the tip of the probe antenna. Finally, the intensity distribution of the focused beam spot was measured by moving the detector using an XYZ translation stage. The bidirectional repeatability of the stage we used is ±1.5 μm.

3. Results and Discussion

The experimental results are shown in Figure 8. The normalized intensity distribution (Figure 8a) by maximum intensity through the fabricated lens confirms that the hot spots repeatedly appear along the direction perpendicular to the lens. The first and second hot spots appear around z = 8 and 16 mm, respectively. The distance between the hot spots is almost the same as the calculated Talbot distance of 7.9 mm, confirming that the proposed lens focuses the THz wave as desired.

The spot size (Figure 8b) along the direction perpendicular to the lens agrees with the calculation results in Figure 3a. Here, the spot size refers to the focal spot area where the intensity remains at least more than 50% of the intensity at the center of the spot. In Figure 8b and Figure 6c, the gray-shaded regions represent the distances where the terahertz wave is not focused, while the non-shaded ones represent the distances at which the focal spots are generated. In the non-shaded regions in Figure 8b, it can be clearly observed that the experimentally obtained size of the hot spots is close to the calculation result. The graph shows that the THz beam is focused at z = 7.5 mm and the spot size is 2.01 mm. However, the THz waves seem to be not well-focused between hot spots, contrary to the calculated results. This is most likely due to the resolution of the probe antenna with a 2.1 mm aperture diameter, which is significantly larger than the spatial resolution (0.01 μm) used in the calculations.

The other noticeable difference between the calculated and experimental results shown in Figure 8b is the very weak intensity between the hot spots of the fabricated SOL. Since this deviation between simulations and actual measurements only occurs at the weak intensity points, this does not pose a major problem when the lens is used for applications, such as product inspection, because only the points of high intensity would be used for measurements.

Figure 8c shows the intensity distribution normalized to the intensity at z = 8 mm, where the red line represents the calculated results, and the blue dots represent the experimental results. The simulations show that the intensity of the first spot is about the same as the intensity of the second spot. However, the experimental results show that the second spot is slightly more intense. This could be due to either a slight lens tilt relative to the incident wave or the intensity distribution of the incident wave. The calculations were performed for an ideal plane wave, but the intensity distribution of the beam used in the experiments is Gaussian.

The DOF is also an important parameter in lens performance. Figure 8c shows that the experimental DOF values (2.3 mm for the first spot and 2.2 mm for the second spot) are more-or-less consistent with the calculation values (2.1 mm for the first spot and 2.45 mm for the second spot).

These results also show that the resolved spot size at around z = 8 mm from the SOL is 2.01 mm (0.67λ). Since the diffraction limit of a conventional lens is 2.46λ, it is demonstrated that the SOL can focus light to about 30% of the spot size by a conventional lens, or 70% beyond the diffraction limit. Figure 8d shows the normalized intensity on x = 0 mm or y = 0 mm at z = 8 mm, and this shows that the spot is circularly focused when viewed in the xy plane.

Thus, with the SOL fabricated for 0.1 THz, we were able to realize a lens that can focus light beyond the diffraction limit while maintaining a depth of focus equivalent to that of a conventional lens [33].

Due to the characteristics of this lens, the focal size and focal length are affected by the frequency of the incident beam. Although our lens is designed for 0.1 THz, a typical oscillator does not emit a single frequency, but rather a certain range of frequencies. Thus, the effect of different incident frequencies was also investigated. Figure 9a,b show the intensity distribution of 0.097 THz and 0.103 THz beams, respectively, after they pass through the lens, as calculated by the finite element method. The distance between focal spots is 7.5 mm in Figure 9a and 8.35 mm in Figure 9b. Figure 9c,d show the experimental intensity distribution of 0.097 THz and 0.103 THz beams, and they agree with the calculation results of Figure 9a,b.

Figure 9e shows the frequency dependence of the interval between focal points. The black line represents the calculated results using the Talbot principle while the red squares show the calculated results by the finite element method. The green circle shows the experimental results. This graph shows good agreement between calculations by the finite element method, experimental results, and calculations by the Talbot principle. This characteristic makes it possible to use the SOL for focusing incident THz beams with different frequencies and consequently performing measurements with variable focal lengths.

4. Conclusions

We designed and successfully fabricated a metamaterial-based 0.1 THz SOL that generates hot spots repeatedly in the direction perpendicular to the lens and experimentally demonstrated its performance. The fabricated lens, with a 101.6 mm diameter and 7.5 mm focal length, overcomes the 2.46λ diffraction limit of conventional lenses by focusing THz radiation onto a small spot of 0.67λ diameter. Moreover, hot spots are formed by each 3875 × 3875 μm² unit pattern composed of five 127.5 μm diameter circular slits. The SOL does not only provide significantly higher resolution than conventional lenses and a working distance that is more than 2λ, it also allows the means to simultaneously focus a THz wave on multiple spots. Furthermore, the SOL can focus incident THz beams with different frequencies and thereby exhibit variable focal lengths, allowing convenient adjustments to the working distance without requiring any lens design changes. Therefore, this lens design is ideal for implementing THz-based industrial inspection applications due to its demonstrated sub-diffraction limit focal spot size, long and adjustable working distance, long depth of field, and multiple hot spots.