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Abstract
Metamaterials with large axial anisotropy posses a nearly flat dispersion profile in k (wave vector) space and thus offer an effective solution to overcome the diffraction limit by supporting the propagating high - k extraordinary modes. However, existing analytical models reveal that resonant high - k slab modes and the polarization dependent ordinary waves cause image distortion in metamaterial slabs. In this paper, we consider a two-dimensional (2D), local, highly anisotropic metamaterial slab as an imaging device and apply a standard transfer matrix approach to calculate the transmission properties of the slab at terahertz (THz) frequencies. Our simple analytical model reveals that resonances induced by the reflections are the main source of deteriorating the image quality, thus requires effective post-processing methods to remove them. For that, we apply an ultra-short super-Gaussian windowing function to minimize the resonant behavior of the metamaterial slabs, observing good imaging over the frequency band of interest. Our numerical method offers a pathway to mitigate observed image artefacts, and are applicable to a range of highly anisotropic metamaterial slabs, e.g., wire metamaterials, layered metamaterials and magnifying hyperlenses. Finally, finite element based software is used to model the 2D metamaterial slab to verify the analytical models.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The resolution of conventional imaging devices is fundamentally constrained by the wave nature of light, commonly know as diffraction limit, which prevents subwavelength features to be transmitted in the far-field [1]. The reason behind this limitation is the loss of finest details of an object contained by the high - $k$ modes, allowing imaging only in the near-field. A practical approach to overcome the diffraction barrier is to use artificial composite materials having extraordinary electromagnetic properties. Metamaterials, in particular, hyperbolic metamaterials are a class of exciting artificial materials, which possess a hyperbolic dispersion profile and offer an unique solution to beat the diffraction limit by supporting the propagating high - $k$ modes, which are evanescent in naturally occurring dielectric mediums [2]. Existing theoretical models of metamaterials show that hyperbolic dispersion can be achieved in many geometrics, but the most easiest and well-studied geometries are the wire array metamaterials [3] and layered metamaterials [4]. The wire metamaterials are comprised of thin parallel metallic wires embedded in a dielectric host [3], whereas the layered metamaterials are made of alternating subwavelength metal-dielectric multilayer stacks [4].
For microwave to several terahertz (THz) frequencies, wire metamaterials posses low loss with a nearly flat iso-frequency contour [2]. Due to the unusual dispersion profile, all high - $k$ modes travel through the medium in the same direction with identical phase, enabling diffraction-free imaging. Owing to low loss, wire metamaterials can transport images over hundreds of wavelengths away from the source plane, making them promising candidate for subdiffraction imaging device [5].
Until now, imaging experiments with wire array metamaterials have been successfully demonstrated at microwave [6], THz [5] and infrared frequencies [7]. Early imaging experiments using wire metamaterials were demonstrated for operation at microwave frequencies, providing subwavelength resolution and capable of transmitting images to several wavelengths away from the source field [6]. In [6], the authors have experimentally verified that good imaging is obtained at a fixed frequency corresponding to the slab length, thus limiting the broadband operation of the device. At THz frequencies, first wire metamaterials were fabricated using the fibre drawing technique [8], a similar fabrication technique to produce photonic crystal fibers [9]. Later, the same group performed the imaging experiments at THz frequencies, showing imaging with resolution and focusing far below the diffraction limit [5]. Though the wire metamaterials proven to be an excellent device for subdiffraction imaging, but their operation is limited by the narrow frequency band obeying the Fabry-Perot resonances (FPRs) of the wire array slab.
Existing analytical models and experiments have shown that optimal imaging is only possible at FPRs, however, any deviation from the FPRs, images are heavily affected by the resonant enhancement of the high - $k$ slab modes and to the excitation of the surface waves supported by the interfaces of the slab [3]. In metamaterial based lenses, a number of techniques have been introduced to avoid the unwanted imaging artefacts. Recently, point defects have been introduced into the array of the metamaterial superlenses that moves the resonant modes into the high spatial frequencies, which leads to avoid imaging artefacts and also reduces the effect of material loss on the subwavelength resolution [10]. In wire based devices, graphene was introduced at both interfaces of the structure, which significantly enhance evanescent waves, ensuring tunable as well as the broadband operation of the device [11]. In wire based tapered magnifying hyperlenses, dielectric phase compensator was introduced to compensate the phase differences along the length of the medium, thus enabling broadband operation of the device [12]. In straight wire based devices, the resonant behavior of the slabs were minimized experimentally [13] as well as with the numerical post-processing methods [14]. Experiments at THz frequencies applied an ultra-short single cycle THz pulses, eliminating all reflected pulses from the main pulse, resulting in a perfect broadband imaging [13]. In the frequency domain, time gating of ultra-short pulse was approximated with spectral convolution [14], providing good imaging across the frequency band of interest. Later, the same group fabricated a prism based magnifying hyperlens [14] and introduced a space dependent convolution technique to get broadband imaging [15]. A similar post-processing technique was applied in metamaterial based hyper-prism for imaging in the far-field [16]. However, one of the main drawbacks of applying the rectangular pulse (sinc in the frequency domain) in [14–16] is that they produce unwanted side lobes in frequency domain.
In this paper, we study the image transmission properties of a simple highly anisotropic metamaterial slab at THz frequencies. We investigate how the resonant high - $k$ slab modes supported by the finite metamaterial lens can affect image quality. Unlike previously reported post-processing methods [13–16], we apply a super-Gaussian windowing function, and using a figure of merit (FOM), we show how imaging is improved over a broad frequency band after applying the proposed method. We also compare our method with the existing one [14], finding improvement in FOM at some frequencies.
2. Theoretical framework
We begin our theoretical analysis by considering the slab as homogeneous, but in the canalization regime [4], the fields only propagate in the direction of optical axis. To simplify our analytical model, we consider the problem to be 2D, local, and highly anisotropic, in which only extraordinary modes propagate through the medium. A 2D schematic of the metamaterial slab under consideration is shown in Fig. 1(a). We consider a slab of length L, which is infinitely extent in the transverse direction, x $\rightarrow$ $\infty$ and is placed at z = 0 to z = L. We assume that the permittivity along the transverse direction is positive ($\varepsilon _{xx}~>~0$), whereas in axial direction the permittivity is negative ($\varepsilon _{zz}~<~0$). Up to several THz frequencies, the value of $\varepsilon _{zz}$ is large and negative, and it’s value can be estimated from homogenization theory, as formulated in [17]. We divide our considered geometry into three regions, as indicated in Fig. 1(a) and apply a transverse magnetic (TM) polarized wave at the input face of the slab. For an obliquely incident TM polarized wave, the magnetic field of the three regions can be expressed as
Fig. 1. (a) A 2D schematic of the highly anisotropic metamaterial slab. The two subwavelength slits are indicated by red color. (b) Simulated propagation of light through a 2 mm long metamaterial slab. The intensity distribution, $|E_x|^{2}$ is taken at 0.3 THz. The vertical black dotted line indicates the location of the exported field.
Fig. 2. (a) 2D map of the transmission coefficient calculated from the local transfer matrix method. (b) Transmission coefficient as a function of normalized spatial frequency, $k_x/k_0$ at 0.28 THz and 0.30 THz. (c) Calculated frequency dependent intensity profile as a function of position and frequency. (d) Calculated intensity profile at 0.28 THz and 0.30 THz. Horizontal black dotted lines in (c) indicate the location of the imaged slits. The top-hat function in (d) is shown in red color.
Figure 2(a) depicts that at all FPRs, unity transmission is observed across the extended $k_x$, and these frequencies offer good subdiffraction imaging, as will be discussed later. As an example, we plot the transmission coefficient as a function of normalized spatial frequency, $k_x/k_0$ at 0.30 THz, which is the fourth FPR corresponding to the slab length, L = 2 mm. As expected, unity transmission is observed (solid red), Fig. 2(b), but at off resonance, taken at 0.28 THz, a strong resonant enhancement of the evanescent modes can be observed in a certain range of $k_x/k_0$. The observed resonant enhancement of the evanescent modes will lead to a severe image distortion as they are amplified by a factor of 10 or more (dotted blue), Fig. 2(b).
We now investigate the imaging performance of the slab for a given subwavelength source field. As a source field we use two top-hat functions with each 200 $\mu$m width and have inner edge separation of 100 $\mu$m. The separation of the slits are below the diffraction limit between the considered frequency range, 0.1 to 0.7 THz. We then calculate the output field by simply applying the inverse Fourier transform of $T(k_x)\Pi (k_x)$, where $\Pi (k_x)$ is the input spatial spectrum calculated by applying the Fourier transform to the top-hat function. The frequency dependent intensity profile as a function of position and frequency (also known as line-scan [5]) is shown in Fig. 2(c). At the location of the slits strong intensity distribution can be observed (horizontal black dotted lines) along with the appearance of periodic side lobes. The spectral periodicity of these inherent side lobes depends on the slab length. We now plot the intensity profile at 0.28 THz and 0.30 THz, as shown in Fig. 2(d). At the FPR, 0.30 THz, the intensity profile matches the location of the source field with no appearance of slide lobes (dotted gray). However, at 0.28 THz (off-resonance), the image is distorted as the side lobes are clearly visible (solid green), justifying that the excitation of resonance induced high - $k$ modes [Fig. 2(b)] are responsible for image distortion.
The 2D local model presented here provides fundamental information of the image transmission properties through metamaterial slabs. We now compare this model with the more realistic wire medium utilizing spatial dispersion [20] and additional boundary conditions [21]. In our analysis we consider the wire medium to be suspended in air, $\varepsilon _{xx}(k_x)$ = 1. The transmission coefficient for the wire medium is calculated from Eq. (9) [20] and is shown in Fig. 3(a). From Fig. 3(a) it can be seen that similar to the 2D local model, the spatially dispersive model also shows flattened periodic arms with near unity transmission bounded over the extended $|k_x/k_0|\leq 1$. The transmission coefficient as a function of normalized spatial frequency, $k_x/k_0$ is plotted at the two frequencies of interest, i.e., at 0.28 THz and 0.30 THz, and is shown in Fig. 3(b). As expected, at the FPR resonance, unity transmission is observed over the normalized spatial frequencies considered, however strong resonant response similar to Fig. 2(b) is also observed at off resonance. Frequency dependent intensity profile for wire medium is depicted in Fig. 3(c), showing similar imaging behaviour as seen for local model, Fig. 2(c). Figure 3(d) shows intensity plot at 0.28 THz and 0.30 THz. The two slits can be well separated when the intensity is plotted at 0.30 THz, whereas side lobes are clearly visible when intensity is taken at 0.28 THz. The simple 2D local model presented here agrees well with the spatially dispersive model utilizing a realistic wire medium.
Fig. 3. (a) 2D map of the transmission coefficient calculated from [20]. (b) Transmission coefficient as a function of normalized spatial frequency, $k_x/k_0$ at 0.28 THz and 0.30 THz. (c) Calculated frequency dependent intensity profile as a function of position and frequency. (d) Calculated intensity profile at 0.28 THz and 0.30 THz. Horizontal black dotted lines in (c) indicate the location of the imaged slits. The top-hat function in (d) is shown in red color.
3. Verification of analytical models
A two-dimensional (2D) finite element method based software (COMSOL) is used to model the propagation of fields through the metamaterial slab and to validate our analytical models. Similar to the analytical models, we use a 2 mm long metamaterial slab hosted in a dielectric with permittivity $\varepsilon _{d}$ = 1. In our simulation, we consider a permittivity tensor [$\varepsilon _{xx}$, $\varepsilon _{zz}$] = [1, -$1000$-5000$i$]. Similar to the analytical models, two subwavelength slits with same dimensions are used as a source field. The field excitation is given by the scattering boundary condition with a plane wave polarized in x direction and the propagation of the wave in z-direction, as indicated in Fig. 1(b). The metamaterial slab considered here is simulated with a triangular mesh with a maximum step width of the mesh size set to 20 $\mu$m. Note, the slits are placed in direct contact with the metamaterial slab to capture the evanescent fields. To block the electromagnetic waves, we use a plasmonic material, silver, a well-studied material at THz frequencies [14]. The permittivity of the silver is then calculated from the well-known Drude model [22]. The simulated transmission of light over the xz - plane is shown in Fig. 1(b). The intensity distribution, |$E_x$|$^2$ is simulated at 0.3 THz. As expected, perfect propagation of light from the source to the image plane is observed. The simulation shows a good illustration to the subwavelength imaging with the metamaterial slab. The exported field region is indicated by the vertical dotted line, Fig. 1(b), which is 50 $\mu$m away from the end of the slab; this distance corresponds to the experiments reported in [5]. Simulated frequency dependent intensity profile of the two imaged slits is shown in Fig. 4(a), and is normalized at each frequency. Similar to analytical results [Figs. 2(c), 3(c)], high intensity central distribution matching the location of the slits can be observed along with the clear appearance of periodic side lobes, Fig. 4(a). As can be seen, image of the slits are well resolved at FPRs, see for example vertical white dotted line, but at off resonance (dotted green), periodic side lobes appear which contribute to the image artefacts. For clarity and to compare with the analytical models, we present images at the two frequencies of interest at 0.28 THz and 0.30 THz, Fig. 4(b). The imaged slits with metamaterial are also compared with input source field. The source field is calculated immediately after the location of the slits, as indicated in Fig. 1(b). Similar to the analytical results, it can be seen that at 0.30 THz, metamaterial slab can clearly separate the two imaged slits, but at 0.28 THz, images are highly distorted.
Fig. 4. (a) Simulated frequency dependent intensity profile as a function of position and frequency. Horizontal black dotted lines indicate the location of the imaged slits. (b) Intensity profile of the imaged slits at 0.28 THz and 0.30 THz. The input source field is shown in red color.
Our simple analytical 2D model as well as the finite element based simulation predicts that imaging with highly anisotropic metamaterials rely on FPRs of the slab, as the resonant high - $k$ slab modes distort the overall image quality. In the following Section, we explain them with the time domain analysis and also present a recipe to overcome the imaging problem discussed here.
4. Post-processing method
Previously reported experiment and numerical methods have demonstrated that time gating of ultra-short THz pulse or it’s equivalent rect function allows separation of the initial pulse from the backreflections. Here we apply an ultra-short super-Gaussian windowing function to avoid the artefacts, a similar post-processing method was recently introduced to remove the resonance effect on Purcell effect in microwave fishnet metamaterial [23]. One of the advantages of the super-Gaussian function is that the shape of the Gaussian function in time domain and the frequency domain is similar, and hence does not have side lobe in frequency domain as of rect function. We have shown that better performance in imaging can be obtained by applying the super-Gaussian function compared to the previously reported method [14], which will be discussed through FOM in the next Section. The super-Gaussian windowing function can be expressed as [23]
where, $\tau$ is the window length of the Gaussian function required to eliminate the subsequent backreflections. The width of the super-Gaussian function can be expressed in terms of slab length and permittivity as $\tau ~=~\sqrt \varepsilon _{xx}L/c +t_l$, where $t_l$ is the width of the excitation THz pulse. In our simulation, the width of the launched THz pulse is taken as 3.5 ps. It is important to note that such a gating width, $\tau$ will not exclude any part of the main signal. To eliminate the side lobes of Fig. 4(a), the window length of the super-Gaussian function should be shorter than the propagation time of the first reflection. In order to obtain the exact location of the initial main pulse and the subsequent reflected pulses, we produce time dependent intensity profile by applying Fourier transform to Fig. 4(a). The temporal intensity profile is shown in Fig. 5(a). Since the length of the metamaterial slab is 2 mm and the medium is considered in air, the arrival time of the first reflected pulse can be calculated from $t_p~=~\frac {3\sqrt \varepsilon _{xx}L}{c}$. The calculated propagation time for the first reflected pulse for 2 mm slab length would be $\approx$20 ps, which is clearly matched with the temporal profile, see the location of the first reflected pulse in Fig. 5(a). The time difference between two reflected pulses can be calculated by $\frac {2\sqrt \varepsilon _{xx}L}{c}$, corresponds to $\approx$13.35 ps, which is also matched with our simulation. By knowing the complete information of the time dependent intensity profile, we can now choose the proper window length of the super-Gaussian function to exclude the reflected pulses.Fig. 5. (a) Simulated temporal intensity profile as a function of position and time. (b) Temporal electric field profile taken at the centre of the upper slit. Inset shows temporal field profile after applying the post-processing method. (c) Frequency dependent intensity profile after applying the super-Gaussian window function. (d) Intensity profile of the imaged slits before processing (dotted green) and after processing (solid blue).
To eliminate the first reflected pulse, we choose width of the window function, $\tau$ = 10 ps, but any value of window length, $\tau$ in the limit $\tau _{1}<\tau \leq t_p$ would be appropriate, where $\tau _{1}$ is the width of main pulse. Figure 5(b) shows simulated temporal electric field profile taken at the centre of the upper slit (solid red). Note, the similar temporal profile can also be obtained by taking at the centre of the lower slit. From Fig. 5(b) it can be clearly seen that in between 20 ps to 50 ps, three reflected pulses are observed, but comparing to the main pulse their amplitude decays with time, as expected. Although the intensity of the reflected pulses are very low, their presence cannot be ignored as they can severely distort the image [15].
The super-Gaussian function with $\tau$ = 10 ps is shown, Fig. 5(b) (solid blue), and the inset shows temporal field profile after applying the super-Gaussian function, eliminating the reflected pulses leaving only the main pulse. Since our initial simulated exported field, $\boldsymbol {E}$ was in frequency domain, we thus convolve $\boldsymbol {E}$ in frequency domain by $\mathfrak {F}$[g(t)]. The simulated frequency dependent intensity profile after applying the windowing function is shown in Fig. 5(c). The frequency dependent artefacts arising from reflections [Fig. 4(c)] have been removed and the images of the two subdiffraction slits are now clearly distinguished over the frequency range of 0.1 to 0.7 THz. As an example, the intensity of the slits at 0.28 THz after convolution (solid blue) is plotted in Fig. 5(d), and the intensity of the unprocessed image (dotted green) is shown at the same frequency for comparison. Noticeable improvement in imaging can be observed compared to the unprocessed image. One of the advantages of applying the post-processing methods is that it provides broadband imaging, meaning the imaging devices do not need to be tuned to the frequencies obeying the FP condition. Note that the width of $\tau$ should be greater than the main pulse, otherwise this leads to a loss of information of the main signal and loss of frequency resolution.
5. Performance analysis using FOM
With a view to comparing the performances of the proposed super-Gaussian function and previously applied rect function, we use a FOM, a quantitative indication of how imaging is improved over a broad frequency band. The FOM can be expressed as [14]
where, $O$ represents electric field distribution of the two perfect rectangular slits and $I$ is the electric field distribution of the imaged slits with metamaterial. We define FOM such that a value of unity is reached if O and I are perfectly matched, i.e., the transmission of the near-field of the slits are perfect. The calculated FOM of the unprocessed data, super-Gaussian function and rect function is shown in Fig. 6. As expected, the FOM of the unprocessed data (solid red) is fluctuating over frequency: the peaks of the red curve indicates the location of the FPRs corresponding good imaging, whereas the troughs of the FOM indicate severe imaging artefacts. After applying the super-Gaussian function (solid blue), one can readily notice that the FOM reaches maximum value with nearly flat in shape, meaning good imaging can be obtained over the entire frequency band of interest. Note that, the improvement of imaging with the windowing function of course comes at a cost of losing frequency resolution and also phase.Fig. 6. FOM of the unprocessed data (solid red), after applying the super-Gaussian function (solid blue), and rect function (dotted green).
We now compare the FOM of the proposed method with method presented in [14]. From Fig. 6 one can notice that the profile of the FOM of the two curves is similar, but applying super-Gaussian windowing function provides arguably better FOM at some frequencies, which may be due to the fact that the super-Gaussian function is free from side lobes in the frequency domain.
6. Conclusion
In conclusion, we have analytically studied the transmission of subwavelength slits through a highly anisotropic metamaterial slab and also verified our findings with the numerical simulation. Our analytical models have demonstrated that resonance induced high - $k$ modes causes image distortion in metamaterial slabs, limiting the broadband operation of the device and thus requires effective strategies to overcome the problem. We have introduced an experimentally feasible post-processing approach to mitigate the observed image artefacts in metamaterial slab, and arguably better than the previously applied approaches. We believe that our method can be applied to all highly anisotropic metamaterials, e.g., wire metamaterials, layered metamaterials and magnifying hyperlenses.
Acknowledgment
The authors would like to thank Dr. Boris Kuhlmey for fruitful discussions. The authors also thank Dr. Alessandro Tuniz for providing initial support in modeling 2D metamaterial slab.
Disclosures
The authors declare no conflicts of interest.
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