1. Introduction

The resolution of conventional imaging systems is limited by diffraction to about half the wavelength of incoming light, which impedes imaging of subwavelength features. These systems cannot transfer subwavelength information since this corresponds to high spatial frequencies, which decay exponentially during propagation in commonly used isotropic materials, and are thus only detectable in the near field [1,2]. Due to the evanescent nature of high spatial frequencies in natural materials of refractive index n, image reconstruction is limited by a resolution of λ/2n at wavelength λ. In order to overcome the diffraction barrier artificial metamaterials may be used, which can have electromagnetic properties beyond those available in nature [2].

Over the past decades a number of proposals have been made to overcome the diffraction limit using metamaterials. A first theoretical approach was the superlens proposed by Pendry [3], which exploited negative refraction. The key concept was that decaying evanescent waves are amplified due to the resonant excitation of surface plasmons at the interfaces of a negative index slab. After several years, the idea was experimentally verified using thin slabs of silver [4,5] and silicon carbide (SiC) [6]. However, it is difficult to create the magnetic properties required for negative index materials [2], and the subwavelength resolving power of most superlenses is strongly reduced by material loss [7], which limits the practical realization of such lenses.

More recently, a new imaging concept was suggested, which used a material possessing a hyperbolic dispersion relation. This so-called hyperlens is extremely anisotropic, possessing metallic behavior in one direction, and dielectric behavior in another. This dispersion relation allows the medium to support high spatial frequency propagating waves, thus subwavelength information propagates rather than evanescently decaying [8]. The first experimental demonstration of imaging devices based on hyperlensing was performed by Liu et al., using layered media consisting of alternating subwavelength layers of metal and dielectric [9], with large-scale fabrication being a major challenge.

Another approach to subwavelength imaging uses wire media [2], i.e., thin metallic wires with subwavelength separation embedded in a dielectric medium. The major advantage of wire media is that they can have low loss [2, 10], as they do not require resonant magnetic properties. From radio to infrared frequencies, wire media operate in the so-called canalization regime [11], where evanescent waves are converted to propagating waves of constant phase velocity. These propagating waves are capable of supporting subwavelength features from the object plane to an image plane at the opposite side of the wire medium, forming a sub-diffraction-limited endoscope. The necessary anisotropy arises from the geometry of the wire media: the axial permittivity (along the wires) ε_zz is negative and very large, whilst the permittivity transverse to the wires is positive.

The first experimental demonstration of subwavelength imaging using wire arrays was reported at GHz frequencies [12], using a manually assembled wire array. Recently, wire array metamaterials were fabricated using fibre drawing methods [13], and magnifying and non-magnifying hyperlenses were demonstrated using this technique at THz frequencies with focusing and resolution up to λ/28 [1]. Such wire arrays operate well at Fabry-Perot resonances (FPRs) of the wire array slab, where transmission amplitude and phase are independent of spatial frequency, leading to high quality images being transmitted to the image plane. However, away from the FPRs images deteriorate due to resonant enhancement of evanescent waves at certain spatial frequencies resulting in image artefacts [14–20]. This limits the use of such wire array materials for subwavelength imaging, as it restricts their ability to be used over wide frequency ranges. More recently [15], it was experimentally demonstrated that the back reflections within the slab which induce artefacts can be filtered out by using ultra-short electromagnetic pulses, resulting in distortion free images between 0.1 and 1.75 THz.

Another source of artefacts are direction-dependent ordinary TE waves which contribute to broadening of the point spread function and may add unwanted background noise and thus affect the image quality [16–20].

In this paper, we numerically investigate several methods for removing all the sources of imaging artefacts mentioned, including methods relying on spectral rather than time-gated ultra-short pulses, which may be more practical. In Section 2 we introduce the wire array endoscope used in this work, and show the presence of the artefacts when using it to image a simple aperture as an example. In Section 3 we introduce three methods to remove such artefacts, which we will refer to as convolution, field averaging, and power averaging, and evaluate their effectiveness and present situations where they may be applied. In Section 4 we apply these methods to more complex apertures and evaluate their effectiveness. Finally in Section 5 we introduce a mathematical projection to filter out ordinary waves, and demonstrate its effectiveness for a range of source images.

2. Modelling of the wire based endoscope

While the methods for improving artefacts for wire array media presented here are quite general, we demonstrate them on the example medium shown in Fig. 1, composed of 121 silver wires of 10 µm diameter and 50 µm pitch arranged in a square pattern, surrounded by air. The wire array is placed 5 µm away from an aperture (object plane), and the image of the aperture is considered on the other side of the array 50 µm away from the end of the wires (image plane). The distance of the image plane from the wire array is chosen to be in close proximity to the wire array to capture evanescent waves, but far enough to not be affected by local edge effects of wires. The wires are directed along the propagation direction (z). The length of the wires is 1 mm, and the dimension of the circular aperture is set to 200 µm, which is smaller than λ/2 for frequencies below 0.7 THz. These dimensions are similar to reported experiments in THz imaging [1].

 

Fig. 1 Geometry of wire array metamaterial. The 3D simulations contain a wire array metamaterial with an 11 × 11 square array of silver wires surrounded by air.

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To calculate the electromagnetic fields, we use a finite integral method based software (CST) as a simulation tool, utilizing two symmetry planes (xz and yz) to reduce the total computation time. Perfect magnetic boundary conditions in the xz-plane, and perfect electric boundary conditions in the yz-plane are included, with a total number of tetrahedral elements greater than 1 million, and consider frequencies from 0.1 to 0.7 THz in 2.5 GHz steps.

The field incident on the aperture is an x-polarised plane wave propagating in the z-direction. The wave diffracts on the aperture, locally generating a monopolar E_x, dipolar E_z and quadropolar E_y distribution, all on the scale of the aperture and thus with a large spread of transverse spatial frequencies [21]. These fields couple to both ordinary and extraordinary waves of the wire medium [16]. In many instances we will consider images only in the xz-plane, in which case the extraordinary, non-diffracting waves are xz polarized. For consistency, in this section and the next we thus only consider the x-component of the electric field at the image plane. Figure 2(a) and (b) show 2D images of the aperture at the image plane at different frequencies, with Fig. 2(a) corresponding to a FPR at 0.15 THz, which results in a good image, while Fig. 2(b) is off-FPR at 0.26 THz, with the image distorted by resonant high spatial frequency slab modes which this work aims to eliminate.

 

Fig. 2 Simulated 2D image of the electric field at (a) 0.15 THz, and (b) 0.26 THz (frequencies shown by arrows on (c)). (c) Simulated electric field intensity as a function of position and frequency at the output of a 1 mm wire array metamaterial with a 200 µm aperture at the input. The dotted lines indicate the edges of the aperture.

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To present the image as a function of frequency more succinctly, the intensity in the middle of the 2D image (corresponding to y = 0) is plotted as a function of x and frequency in Fig. 2(c), which will be referred to as a line-scan, which is then normalized at each frequency. Figure 2(c) shows that the 200 µm aperture is well resolved at all FPRs, but strong periodic side lobes appear away from the FPRs. In the time domain, the artefacts can be seen as arising from multiple reflections along the length of the endoscope constructively interfering [15]. This is clear in the time evolution of the field in the lens (see Fig. 3 and Visualization 1), which shows the reflection contributing to the side lobes, forming standing waves. Since the length of the wires is 1mm, the spectral periodicity of the side lobes corresponds to the FPR spacing of 150 GHz.

 

Fig. 3 Frame from Visualization 1: Time evolution of the electric field through 1 mm hyperlens (single-frame).

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When time resolved measurements are available, filtering out the first pulse before any additional reflections can remove the side lobes [15]. This time-filtering, successfully demonstrated using time domain THz spectroscopy experiments, is equivalent to a convolution with a sinc function in the Fourier domain, which can be done if both phase and amplitude of the signal are available. At higher frequencies than THz it is rare to have experimental access to a time-domain signal with resolution sufficient to isolate individual reflections, or alternatively a complete set of amplitude and phase signal at all frequencies, motivating the exploration of alternative methods to remove side lobes discussed in Sections 3 and 4.

Even at the FPR, the image takes an oval shape, spreading in the y direction [Fig. 2(a)]. This results from the fact that x-polarized waves excite extraordinary, non-diffracting waves for wave-vectors in the xz- plane, but ordinary, diffracting waves in the y direction [1]. We will discuss how to correct this distortion in Section 5.

3. Removing image artefacts

Previous work has shown that time gating of ultra-short pulsed measurements can effectively remove the reflections which induce the artefacts [15]. This is the first method we consider here, albeit in the frequency domain. We also consider methods that approximate the convolution, through field averaging (which mathematically is a convolution with a rectangular function) and power averaging, where the electric field or power are averaged over full spectral range (frequency spacing between two resonances, Δf = c/2L), respectively. By applying these methods, we show that imaging artefacts can be removed or at least reduced for wide frequency bands, increasing the frequency range over which high quality, subwavelength resolution images may be obtained. From a practical point of view, the first two methods have significance when phase and amplitude can be measured, which is typically the case for terahertz frequencies and below. At higher frequencies, it is more common to measure the intensity of the electric field without the phase. Intensity averaging can then be done either numerically based on multiple narrow band measurements, or more readily using an optical bandpass filter with appropriate bandwidth.

3.1. Convolution

The first method emulates time gating but in the frequency domain. The time gated field $e^{\prime }$is

Here, $\tau$is the gate width required to eliminate reflections from which the side lobes emerge, which should be longer than the sum of pulse length and simple propagation through the medium, but shorter than the propagation time of the first reflection, corresponding to 3nL/c. Since multiplication in the time domain is equivalent to a convolution in the frequency domain, we take the Fourier transform of Eq. (1) to obtain

Note that throughout this work, temporal Fourier transforms are expressed using upper case letters. To determine an appropriate gate width, the time dependent electric field intensity equivalent to Fig. 2(c) was obtained through a Fourier transform, and is shown in Fig. 4(a).

 

Fig. 4 (a) Simulated electric field intensity as a function of position and time. Simulated electric field intensity after convolution for different widths of the gate pulse (b) 8 ps (c) 12 ps, and (d) 50 ps. (e) Simulated intensity profile of the bare aperture. The output is taken 50 µm away from the aperture. Simulated 2D image of the electric field at (f) 0.26 THz, and (g) after convolution at 0.26 THz.

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This reveals that the radiation forming the image arrives over a time of 8 ps, corresponding to the width of the launched pulse and the time for a single pass through the lens, L/c. Reflected pulses are separated by 2L/c = 6.7 ps. In order to avoid the first reflection we choose τ = 8 ps, noting that any value τ between 8 ps and the arrival time of the next reflected pulse 3L/c = 10 ps would be appropriate. This choice of τ corresponds to sinc function with full width half maximum of 150 GHz, and (somewhat coincidentally) also corresponds to one free spectral range of the structure.

Figure 4(b) shows the line-scan of Fig. 2(c) after convolution. The periodic side lobes completely vanish over the frequency range considered, showing remarkable improvement over Fig. 2(c), making this method promising for imaging applications. A shorter gate width, (e.g., 1.5 ps) corresponding to a wider sinc function in the frequency domain, results in loss of information of the main signal and loss of frequency resolution. A longer gate width, (in this case, 12 ps and 50 ps), reveals that multiple reflections are responsible for the periodic side lobes as shown in Figs. 4(c) and (d). The 2D image of the electric field away from the FPR at 0.26 THz before [Fig. 4(f), as in Fig. 2(b)] and after convolution [Fig. 4(g)] then show that the side lobes completely vanish, leading to a significant improvement in imaging.

3.2. Field average

The second method we consider is electric field averaging over a finite spectral width. At terahertz and lower frequencies, the amplitude and phase of the electric filed can typically be measured, and the complex electric field is simply averaged over a given frequency range. We illustrate this method using similar filtering widths as the sinc function of the previous section, namely half a spectral range and a full spectral range of the wire array.

Simulation results show that after averaging, the strong periodic side lobes are reduced in intensity, resulting in better images over the entire frequency range considered. The line-scans shown in Fig. 5(a) and (b) show that at some frequencies the side lobes are reduced to a level of −13 dB, from −2 dB prior to averaging [Fig. 2 (c)]. The 2D images in Fig. 5(c), again away from resonance at 0.26 THz, show that after averaging the field, the side lobes vanish, leading to a much improved image. The quality of the image was found to be rather insensitive to the frequency range over which the field was averaged, however better results are obtained when averaging over a wider frequency range, at the expense of frequency resolution. It is thus preferable to average the field over a region between half- and full- spectral range, so as to minimize the artefacts without losing frequency resolution.

 

Fig. 5 Simulated electric field as a function of position and frequency after averaging electric field over (a) half a spectral range, and (b) full spectral range. (c) Simulated 2D image of the electric field after averaging field over full spectral range at 0.26 THz.

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3.3. Power average

The third method of averaging the power is practical when only the intensity (and not the phase) is measured, e.g., at visible frequencies. As shown in Figs. 6(a) and (b) after averaging the power over half a spectral range and a full spectral range respectively, the intensity of the side lobes decreases, however the images become slightly blurred at frequencies corresponding to FPRs. The intensity of the side lobes reduces to −7 dB at frequencies above 0.4 THz, resulting in better images than without averaging, as confirmed by the 2D image at 0.26 THz [Fig. 6(c)]. However, at FPRs image blurring is more significant.

 

Fig. 6 Simulated power average as a function of position and frequency over (a) half a spectral range, and (b) full spectral range. (c) Simulated 2D image of the electric field after averaging power over full spectral range at 0.26 THz.

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3.4. Figure of merit

In order to compare the performance of the above methods, a figure of merit (FOM) is defined as

 

Fig. 7 FOM (Eq. (4) for different methods as a function of frequency. Arrows indicate Fabry-Perot resonances.

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4. Application to complex apertures

We now apply the same methods to a double subwavelength aperture. For the same wire array, we now consider two 200 μm diameter apertures, with inner edge-to-edge separation of 100 μm. The results are shown in Fig. 8. Figure 8 (a) shows that the wire array metamaterial clearly resolves double apertures at the FPRs, however severe artefacts are present, possibly due to additional contributions from charge coupling between apertures [1]. As a consequence, for frequencies far below 0.4 THz it is difficult to distinguish the apertures over broad frequency bands, while above 0.4 THz there are two strong central intensity distributions matching the location of the apertures. Using convolution [Fig. 8(b)] and field averaging [Fig. 8(c)] allows us to resolve the subwavelength apertures with fewer artefacts. Power averaging [Fig. 8(d)] does not perform well below 0.4 THz. The 2D images of the field at 0.26 THz are also shown. Figure 8(e) shows that at 0.26 THz, the two apertures cannot be distinguished using the wire array metamaterial alone. Using convolution and field averaging methods in Figs. 8(f) and (g), the two subwavelength apertures become clearly visible, whereas averaging power [Fig. 8(h)], while arguably better than the raw image, still suffers from considerable artefacts – in particular with a prominent central lobe between the location of the two apertures.

 

Fig. 8 (a) Simulated line-scan of the electric field of two 200 µm diameter apertures with 100 µm inner-edge separation as a function of position and frequency. Simulated line-scan after (b) convolving field with the sinc function, (c) field average, and (d) power average over a full spectral range. Simulated 2D image (e) unprocessed, and after (f) convolution, (g) field average, and (h) power average at 0.26 THz. Dotted lines indicate location of the apertures (top row).

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As a second example we consider a “V”-shaped aperture as shown in Fig. 9(a). The wire medium shown here contains 12 × 13 squarely arranged silver wires of 1 mm length. The length of the tilted arm of the “V” is 365 µm and the width of the arm is 150 µm. We first image the field at frequencies corresponding to a FPR and off-resonance, i.e., at 0.30 THz, and at 0.26 THz. After the wire array we see that at 0.30 THz, the image can be resolved but away from the resonance the image is distorted, as expected. Applying the aforementioned methods, imaging of such a complex aperture is possible away from FPRs. Again convolution and field averaging provide considerable improvements in image quality, with power averaging also yielding better images than the raw output, albeit with relatively strong remaining artefacts.

 

Fig. 9 (a) Schematic of 1 mm wire array metamaterial with “V” shape aperture. Simulated 2D image of the electric field with the metamaterial fibre at (b) 0.30 THz, and (c) 0.26 THz. Simulated 2D image of the field at 0.26 THz of (d) the bare aperture, and the aperture with the wire array (e) after convolution, (f) field averaging, and (g) power averaging.

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5. Removing ordinary waves using Fourier space projection

For wire media in the canalization regime, images are formed through a slab because the TM modes (or quasi TEM) modes have flat isofrequency contours, with all spatial frequencies propagating with identical phase- and group- velocity. The ordinary TE modes however have circular frequency contours identical to those of waves in isotropic media, and thus diffract, but only exist for low spatial frequencies. When ordinary waves are excited, they contribute to additional blurriness or background noise to the image. In many hyperlens studies so far, the role of ordinary waves was overlooked, as typical objects studied were small compared to the wavelength: small objects do not have strong low spatial frequencies and thus cannot excite TE modes effectively. For objects larger than the wavelength, diffracting TE waves are excited, adding low spatial frequency noise and additional distortion [16]. TE and TM waves differ in their polarization, and can thus in principle be segregated: the electric field of TE waves is orthogonal to$k$and$\widehat{z}$, while the electric field of TM waves is in the $(k,\widehat{z})$plane [Fig. 10]. Since only the TM mode has a longitudinal component, segregation between TE and TM mode is straightforward if one is able to measure$E_z$ [16]. However, in practice transverse polarizations are more readily measured. In one-dimensional images TE and TM waves can simply be separated using a transverse linear polarizer, but for two-dimensional images the polarization of TE and TM modes depends on the direction of$k$. Polarization filtering then needs to be done in the Fourier space, and requires measuring both the amplitude and phase of both the transverse field components,$E_x$and$E_y$ which together form the transverse field$E_t(x,y)$. Let ${\tilde{E}}_t(k_t)$be its spatial Fourier transform in the $(x,y)$measurement plane, where$k_t=k_x\widehat{x}+k_y\widehat{y}$. In the Fourier domain, the ordinary$({\tilde{E}}_o)$and extraordinary$({\tilde{E}}_e)$wave contributions to the transverse field can be separated through projection, since${\tilde{E}}_o(k_t).k_t=0$, and${\tilde{E}}_e(k_t).k_t\ne 0$, leading to

 

Fig. 10 Ordinary waves (blue line) and extraordinary waves (red line) in Fourier space.

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Figure 11 summarizes the simulated 2D images of a single aperture, a double aperture, and a larger more complex aperture (“THz” letters, as a proof-of-concept), as obtained from the power output ${\left|E(x,y)\right|}^2$(left column) and the projected power ${\left|E_e(x,y)\right|}^2$(middle column).

 

Fig. 11 Normalized 2D images of power output, projected power and contribution of ordinary waves (left to right). Simulated 2D images of: Single aperture through a wire medium (top row), double aperture through a wire medium (second row), and double aperture without wire medium (third row), with dimensions and details described in the article. Bottom row: 2D images of a larger object imaged through a slab of hyperbolic medium of thickness 3.4 mm (bottom row). For single aperture and double aperture images are taken at 1.5 THz, while for the “THz” letters, images are taken at 0.58 THz. First three rows calculated using full CST simulations of a wire array, bottom row calculated using a transfer matrix method for hyperbolic media, with material parameters identical to those in Ref. 16. The maximum intensity of the extraordinary waves for single aperture, double aperture (with wire medium), double aperture (without wire medium), and “THz” letter are 11, 4, 4.8, and 4 times higher than that of ordinary waves respectively.

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The contribution of ordinary waves $(E_o=E_t-E_e)$to the total power is shown in the last column. The top row is for a single aperture of diameter 300μm, with fields taken 50 µm away from the end of the wire array, at 1.5 THz. The higher frequency (shorter wavelength) is chosen so that ordinary waves are excited. Since the excitation wave is x-polarized, the field at the aperture is also predominantly x-polarized and the strongest contribution to ordinary TE waves is in the y direction. This manifests as additional blurriness/noise in the y-direction in the top left image of Fig. 11. These artefacts are clearly removed in the image as extracted using Eq. (5) (middle column), leading to a much higher quality image.

The contribution of ordinary waves (last column) is consistent with predominant excitation of wave vectors in the y direction and the spreading of diffractive waves. The second row shows simulated images for a double aperture, each 300 µm in diameter and with inner edge separation 100 µm. Again the image obtained through projection on extraordinary waves is considerably improved compared to the raw image in that one can now recognize the shape of the apertures. However, the projected field distribution has an unexpected maximum in-between the two apertures, which appears to be an artefact introduced by the projection. To understand the origin of this apparent artefact we simulate the same double aperture without the wire array, and decompose the fields using the same projection – as seen in the third row of Fig. 11, now plotted in a logarithmic color scale to accentuate the low intensity details. The intensity peak between apertures clearly already exists in between the apertures in the full field, but is relatively weak. The field components corresponding to extraordinary waves (middle column) accentuate this detail, which is then reflected in the final image through the wire medium.

Finally, we demonstrate the effectiveness of this “projection method” on a larger object with more intricate details, consisting of the letters “THz” using the same method and parameters as in [16]. This simulation uses a transfer matrix method based on homogenized metamaterial properties, as a full simulation of wire arrays of this size is computationally impractical. The electric fields are taken on the surface of the slab at 0.58 THz, and the object consists of dipole point sources at a distance of λ/50 from the hyperbolic material slab [16]. Strong contributions of ordinary waves for the larger “THz” letters badly affect the quality of image (bottom row, left). After applying the projection method the artefacts caused by ordinary waves are entirely removed, resulting in a remarkable improvement in image quality.

6. Conclusion

Endoscopes and hyperlenses based on wire media can resolve subwavelength details far below the diffraction limit, but images from them are not perfect. We have seen that much better images are obtained at FPRs. Away from FPRs, images deteriorate due to internal reflections within the wire media, with ordinary waves degrading image quality further. Here, we developed post-processing methods in the frequency domain to eliminate the unwanted artefacts. Since artefacts arise from multiple reflections within the wire array, they can be eliminated if the main pulse can be separated in time from the subsequent internal reflections. In the frequency domain this can be reproduced by a convolution, or approximated by spectrally averaging fields and power, all improving image quality away from FPRs. Spectral averaging of intensity is equivalent to taking images through a band-pass filter and thus correspond to practical solutions not relying on phase measurements.

This improvement in images of course comes at a cost of losing frequency resolution, and also phase. Although these methods do not improve imaging at the narrow FPR frequency, such imaging is often impractical. Our work shows that even if one is not able to resolve the resonance exactly, a spectral average of field or intensity can still provide good imaging performance. This is of particular relevance to long wire media endoscopes where the free spectral range can be very small, and frequency averaged measurements are inherent.

The other main cause of image deterioration in devices relying on hyperbolic media is the existence of ordinary waves. These can be eliminated when measuring the longitudinal near field, but this is often impractical. We showed that images taken with transverse polarizations only can be projected onto the basis of extraordinary waves in the spatial Fourier plane, yielding high quality images without ordinary wave artefacts. This method could be extended to far field measurements of images magnified by hyperlens [8].

Our work relied on simulations of wire media, which are strongly hyperbolic media in the canalization regime, and are the most promising implementation for hyperlenses at GHz, THz and far infrared frequencies. At visible frequencies, other forms of hyperbolic media (e.g., layered media) may be more practical, and do not operate in the canalization regime. In these cases, different transverse spatial frequencies have different phase and group velocities, and the spectral convolution and averaging methods will need to be adjusted accordingly. The separation of ordinary and extraordinary waves using projection in Fourier space however can still be applied directly.