1. Introduction

Far-field imaging beyond the diffraction limit is a long sought-after goal across various imaging applications in the microwave, terahertz, optical and ultrasound regimes. One of the key issues for this challenging topic is to exploit, whether directly or indirectly, the evanescent waves emerged from the object under consideration [1–5]. For instance, the super-oscillation imaging is a useful technique which relies on the strong coherence between the evanescent waves and propagating waves for the objects fallen into a very narrow field of view (FOV) [3]. The alternative approach is to introduce a cluster of scatters in the vicinity of the probed objects for converting the evanescent waves into propagating waves, leading to an image of the probed object with the subwavelength resolution [4, 5].

Conventionally, there are two types of popular active imaging systems for data acquisition: the real-aperture (RA) system and synthetic-aperture (SA) system. The SA system relies on the mechanical movement of a single sensor to form virtually a large scanning aperture via post-processing, which is typically inefficient in data acquisition [6]. On the contrary, the RA system is composed of a large number of antenna elements, which has more flexibility in measurement modes, but sacrifices the size, weight, power, and price advantages of the single sensor system [7]. Now, a natural question arises: is it faithful to get a subwavelength image from a single sensor with fixed location? The answer is encouraging [8–12]. In [8], Pierrat et al. investigated theoretically the feasibility of subwavelength focusing of a single broadband dipole source in an open disordered medium with a single antenna. Inspired by the theory of compressive sensing (CS), Hunt et al. [9] and Lipworth et al. [10] developed the meta-imager consisting of a single antenna with multiple resonant frequencies, along with a sparsity-promoted nonlinear solver, by which a high-resolution imaging of the scene with sparse objects is achievable.

This paper investigates the super-resolution imaging of soft-matter objects using a single sensor, which benefits from the use of spatio-temporal resonant aperture and sparse reconstruction. A set of proof-of-concept investigations has been performed to verify the proposed theory. The rest of this paper is organized as follows. Section 2 provides a theoretical analysis to provide the physical insights into the proposed methodology of far-field imaging beyond the diffraction limit. Section 3 gives a selected number of numerical results to verify the proposed theory. Finally, section 4 summarize this work.

2. Methodology

This work proposes a novel concept of subwavelength imaging using a single sensor in combination with a spatio-temporal resonant aperture, where the resonant aperture is placed in the vicinity of probed object, and the single sensor is far from both resonant aperture and probed object, as sketched in Fig. 1. The resonant aperture consists of a homogeneous square aperture with finite extension $L_x\times L_y$ and thickness d, which is filled by the dispersive metamaterial with the relative permittivity described by the Drude model${\epsilon }_r=1-\frac{{\omega }_p^2}{{\omega }^2-i\omega \Gamma }$, where the bulk plasmon frequency${\omega }_p=1.62\times {10}^{10}rad/s$, and the collision frequency$\Gamma =7.64\times {10}^{7}rad/s$ are considered in this work. In the following discussions, we demonstrate that such resonant aperture manages the conversion of evanescent waves into propagating waves, and hence the proposed imaging system is capable of producing super-resolution imaging without using any near-field scanning, mechanical movement, and antenna array.

 

Fig. 1 (a) The schematic setup of proposed imaging system for the far-field imaging beyond the subwavelength resolution, which consists of a single far-field sensor located at $r_d=(x,y,z)$and a spatio-temporal resonant aperture in the vicinity of the probed object. Here, the object under consideration is a Chinese character “北”. (b) The schematic configuration of the resonant aperture, which is a square hole in a perfect conductor filled with a dispersive material characterized by the Drude model${\epsilon }_r=1-\frac{{\omega }_p^2}{{\omega }^2-i\omega \Gamma }$, where${\omega }_p=1.62\times {10}^{10}rad/s$ and $\Gamma =7.64\times {10}^{7}rad/s$.

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2.1 Relation to the SPP scattering

This section presents a theoretical analysis to provide the physical insights into the proposed methodology of far-field imaging beyond the diffraction limit. Assuming that the resonant aperture is illuminated by a transverse-magnetic (TM) polarized plane wave$E_{inc}=E_0{e}^{i{k}_{in}\cdot r}$, where$k_{in}=(k_{in,x},k_{in,y},k_{in,z})$, $k_{in,z}=\sqrt{k_0^2-k_{in,x}^2-k_{in,y}^2}$, and$k_0$ is the free-space wavenumber. Note that the case of $k_{in,\rho }=\sqrt{k_{in,x}^2+k_{in,y}^2}>k_0$ corresponds to the illumination of evanescent waves, and $k_{in,\rho }<k_0$ for propagating wave.For simplicity, this resonant aperture is modeled in a local sense, which means that the transmission field at z = d behaviors as, for instance, for the TM case, $E\propto T^{TM}{e}^{i{k}_{iz}d}{e}^{i{k}_{inc,\perp }\cdot r_{\perp }}$for$r_{\perp }\in \Omega$, otherwise being zero. Of course, more accurate analysis can be made by via a full-wave solver of the coupled-mode method, for instance [13, 14].In light of the vector Huygens’ diffraction principle, the electrical field perpendicular to the z direction at far field, $r_d=(\rho ,z)$, amounts to be (see Appendix A for details):

A set of simulations are performed to verify the above claims. Figure 2 shows the dependence of the x-polarized electric field acquired at $r_d$ on $k_{in,\rho }/k_p$ ($k_p=2\pi /{\omega }_p$, and $k_{p}d=0.25$ is considered below) and$\omega /{\omega }_p$, which is calculated using Eq. (1). From this figure, we can see that the illumination of evanescent waves can be efficiently captured by the sensor at far fields, implying that the evanescent waves have been converted into propagating waves after experiencing the resonant aperture, as illustrated by the light region. Furthermore, the light region is associated with the dispersive property of the resonant aperture filled by the Drude medium, i.e.,

 

Fig. 2 The amplitudes of x-polarized electric field as the function of $k_{in,\rho }/k_p$ and $\omega /{\omega }_p$acquired at$r_d=(0,2.5{\lambda }_0,5{\lambda }_0)$, where the incidence is a TM-polarized plane wave, the x-axis is$k_{in,\rho }/k_p$, and the y-axis denotes $\omega /{\omega }_p$. This result is calculated by Eq. (1).

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Now, we can confirm that part of evanescent waves incident to the finite resonant aperture are converted into propagating waves. To show this more clearly, we make the following arguments. For an infinite homogeneous film, its eigenmodes are usually taken as $e^{i{k}_{x}x}$ whether inside or outside the film, which implies that there is no conversion between the evanescent and the propagating waves. However, the eigenfunction is not $e^{i{k}_{x}x}$ any more when the film is truncated into a finite size in the lateral dimension, which means that the evanescent waves and propagating waves are now coupled inside the resonant aperture, as illustrated by the coupled-mode theory [13, 14]. More importantly, the propagating waves converted from the evanescent illumination can be significantly enhanced by the strong Fabry-Perot resonance happened in the resonant aperture filled with dispersive materials. In order to see this claim, a set of numerical simulations are conducted in the context of coupled-mode theory. Assuming that the y-component of magnetic field in the region of z>d is $H_y(r)={\displaystyle \sum }_{m=0}^{\infty }{E}_m^O{G}_m(r)$, where$G_m(r)$is the free-space Green’s function $G\left(r,r^{\text{'}}\right)$projected into the mth aperture mode.The magnetic field emerged from the resonant aperture can be represented by [13]:

 

Fig. 3 The map of $PS=\sqrt{{\displaystyle {\sum }_{m=0}^{Int(2L_x/\lambda )}|E_m^O{|}^2}}$normalized by its maximum as the function of $k_{in,\rho }/k_p$ and$\omega /{\omega }_p$, where simulation parameters are same as those considered in Fig. 2. The Matlab code of reproducing this result can be freely achieved by sending a request email to lianlin.li@pku.edu.cn.

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Above discussions show that a key point for the underlying problem is to find an efficient way to convert evanescent waves arising from the probed objects to the propagating waves. It is well accepted that a finite-size aperture is capable of converting parts of evanescent waves emerging from the imaged objects into the propagating waves. Now, the key point is how to make the conversion efficiency as high as possible. One of feasible solutions to this problem is based on the resonance happened inside the aperture.For the aperture filled with some material, there are two mechanisms of resonances: one is due to the aperture itself, and the other from the structure of “air- medium-air”. Of course, the coupling between these two mechanisms is also possible.To our best knowledge, the first mechanism is problematic. It was observed that the larger the single aperture size is, the weaker the resonance is [13]. Moreover, the resonance is very weak and can be ignorable when the aperture size is comparable to the operational wavelength. For the second mechanism, the resonance is related to the SPPs, where the filled material should be with negative permittivity in order to substantially enhance this conversion. Furthermore, according to the dispersive curve of the air-dispersive medium-air structure, roughly, there is one-to-one relation between $k_{in,\rho }$ and operational frequency$\omega$, where $k_{in,\rho }$ ($k_{in,\rho }>k_0$) denotes the transverse components of wavenumber. However, if the filled medium is non-dispersive, there is only one solution to the resonance equation, which implies that only the evanescent component with $k_{in,\rho }$being the solution to the resonance equation can be efficiently converted into propagating waves. For the purpose of subwavelength imaging, and that the dispersive medium is required to convert as much evanescent waves as possible into the propagating waves. To see above discussions clearly, we conduct two sets of simulations as done in Fig. 3. Figure 4(a) is for the filled materials with ${\epsilon }_r$ = 8, and Fig. 4(b)for ${\epsilon }_r$ = −2.From Fig. 4, we can see that the non-dispersive medium cannot be used to enhance sufficiently the efficiency of converting the evanescent waves into propagating waves. Note that there is a jump at $\frac{\omega }{{\omega }_p}=0.5$, which can be explained that a guide mode (m = 1)changes its character from evanescent to propagating [13].

 

Fig. 4 The map of normalized PS as a function of $k_{in,\rho }/k_p$ and $\omega /{\omega }_p$ when the radiation aperture is filled with homogeneous non-dispersive medium with relative permittivity ${\epsilon }_r$. (a) is for the case of ${\epsilon }_r$ = 8, (b) for ${\epsilon }_r$ = 2,.This figures are normalized by their own maximum.

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2.2 Relation to the information capacity

An essential point of improving the resolution of an imaging system is to enhance its information capacity, which is usually represented by the temporal-bandwidth product. Actually, the information capacity of a broadband imaging system is of the order $O(T\times B)$ [12,17], herein B is the temporal bandwidth and T is duration of time-domain response. Apparently, for the fixed operational bandwidth, only one way of enhancing the information capacity is to make T as big as possible. Moreover, from Eq. (1), we can derive the sensitivity of the frequency-dependent measurements with respect to frequencies, namely,

To see it more clearly, we perform a set of numerical simulations using the CST Microwave Studio 2012 package with the Transient solver. We excite the resonant aperture with an ideal electrical dipole source placed at 18.76mm from the bottom of resonant aperture,which is fulfilled by a small x-aligned CST discrete port with size of 7.5mm. The excitation signal is a narrow pulse with 2.5ns duration (see the red line in Fig. 5(a)). The far-field is probed with the CST far-field probe placed at $r_d$, and both its frequency-dependent and time-dependent responses are shown by the black lines in 5(a) and 5(b), respectively. This set of figures demonstrates that a narrow pulse will show rich peaks in the frequency domain after experiencing the resonant aperture; accordingly, it will be fully expanded in the time domain with a factor of more than 100. Consequently, from the respect of the information capacity, the degree of freedom of the measurements will be remarkably driven up with a factor of 100 and beyond.

 

Fig. 5 (a) A small dipole source of 2.5ns (red line), which is placed at z = −18.76mm away from the bottom of the resonant aperture (${\lambda }_0=188nm$). The response at $r_d=(0,94mm,62mm)$ is recorded and shown by the black line. The inset is the zoomed part. (b) The normalized amplitude of the frequency-dependent response at$r_d$. This set of figures shows many abrupt changes within a very small frequency separation, implying that the response is highly sensitivity to frequency.

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We also emphasize that the near-field coupling between the imaged objects and resonant aperture encodes subwavelength details of objects into the temporal measurements in the far-field region. Hence, in order to obtain the subwavelength imaging from far-field measurements by a single broadband radar, we conclude that three basic requirements should be satisfied: the strong spatial-temporal resonant aperture antenna, near-field coupling, and temporal (or broadband) illumination.

3. Simulation results

Based on above principles, we propose the far-field imaging beyond the diffraction limit using a single sensor in combination with a resonant aperture, which is verified by full-wave numerical simulations. Here, the simulation parameters are set as: $r_d=(5{\lambda }_0,5{\lambda }_0,10{\lambda }_0)$,and the operational wavelength varies from 150 mm to 200 mm with a step of 0.02 mmfor numerical implementation of imaging algorithm. In addition, the object under consideration is placed at $0.1{\lambda }_0$ from the bottom of the resonant aperture. The probed object consists of a Chinese character “北”, with a refraction index of 2.7, as illustrated in Fig. 6(a). The distance between the centers of the two neighbored objects is set to be 12 mm. Note that we voluntarily opt for a low-refractive-index-contrast, which is typical for soft-matter objects, and has ignorable effects on the resonance property of the resonant aperture. In addition, the simulation data input of full-vectorial electric field to the reconstruction procedure is generated by using CST Microwave Studio 2012.

 

Fig. 6 Ground truth and reconstructed results by different methods. (a)The ground truth consisting of a Chinese character “北”. (b)Reconstruction results for different noise level of 40dB, 30dB, and 20dB using proposed imaging system as sketched in Fig. 1(a). In the simulation, the single sensor is located at$r_d=(5{\lambda }_0,5{\lambda }_0,10{\lambda }_0)$, the wavelengths used for broadband illumination varies from 150mm to 300 mm with a step of 0.02 mm, and the probed objects is located at 0.1${\lambda }_0$ at the bottom of the resonant aperture antenna. (c)Reconstruction results without the resonant aperture antenna, where other simulation parameters are the same as those used in Fig. 5(b) but SNR being 30dB. (d)Reconstruction result without the resonant aperture antenna, where $50\times 50$ receivers are uniformly distributed over a square of $10{\lambda }_0\times 10{\lambda }_0$ at$z=10{\lambda }_0$, and SNR is set to be 30dB.

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Basically, the frequency-dependent measurement data$E(\omega )$ acquired at $r_d$is related to the probed object characterized by its contrast$O(r)=\frac{{\epsilon }_r(r)-1}{4\pi }$ (${\epsilon }_r(r)$denotes the relative permittivity of probed object) by$E(\omega )={\displaystyle {\int }_{D}G(r_d,r\text{'};\omega )E_{in}(r\text{'};\omega )O(r\text{'})dr\text{'}}$, where $E_{in}(r\text{'};\omega )$ is the incidence field for the presence of the resonant aperture, and the integral is performed over the region of interest. Herein,$G(r_d,r\text{'};\omega )$ denotes the Green’s function relating a point source at$r\text{'}$and its response at$r_d$. Note that both$G(r_d,r\text{'};\omega )$ and $E_{in}(r\text{'};\omega )$ are known to be a prior. Regarding the inverse procedure, the object can be retrieved from the frequency-dependent measurements$E(\omega )$ by optimizing the following cost function, i.e.,

To mimic the realistic experimental environment, we would like to investigate the robust of proposed methodology to the uncertainty in the illumination source, such as phase/amplitudeerrors, secondary illumination, diffraction, and other practical sources of error. To that end, assuming the ideal illumination as $E_{in}$, then the uncertainty illumination is generated according to $E_{in\_un}=\text{Re}\left(E_{in}\right)\times \left(1+\sigma \times rand\text{n}\right)+i \text{Im}\left(E_{in}\right)\times \left(1+\sigma \times randn\right)$, herein $i=\sqrt{-1}$, $\text{Re}\left(E_{in}\right)$ and $\text{Im}\left(E_{in}\right)$ mean respectively the real and imaginary parts of $E_{in}$, and $randn$ means the generator of Gaussian random number with zero mean and unit variance. Note that we independently perturbs the real part and imaginary part of $E_{in}$, which means there is uncertainty on both phase and amplitude of the illumination. A set of simulations are performed for different levels of uncertainty of phase/amplitude in the illumination sources. The results are shown below, where the SNR for the additive white noise is also added, and set to be 30dB. It can be observed from Fig. 7 that, compared with Fig. 6(b), there are some salt spot noises in the reconstruction images, which can be explained by the multiplicative noise rather than the additive noise. Nonetheless, we can see the proposed method is robust to the uncertainty in the illumination source.

 

Fig. 7 This set of figures are for some uncertainty on the phase/amplitude of plane wave illumination. (a) $\sigma =0.01$,(b) $\sigma =0.03$, and (c) $\sigma =0.05$. For these results, 30dB additive white noise is considered. Additionally, the values of regularization factors are set to be 1, 10 and 50, respectively. Other simulation parameters are the same as those used in Fig. 6.

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4. Summary

In summary, we develop a novel concept for super-resolution imaging using a single sensor, which benefits from the use of spatio-temporal resonant aperture and sparse reconstruction. We have shown that for a given finite operational bandwidth, the resonant aperture antenna can increase the information capacity of measurements efficiently in far fields due to strong Fabry-Perot resonance, leading to the far-field imaging beyond the diffraction limit. A set of proof-of-concept investigations has been performed to verify the proposed theory. It is expected that such imaging concept can find real applications for subwavelength imaging by using more specialized spatial-temporal aperture and efficient reconstruction solver. In addition, the proposed imaging system amounts to encode the spatial details of the probed targets into temporal domain, which mathematically leads to a measurement matrix well matched to CS, and serves as an easy-implementation apparatus for compressive measurements [18–20].

Appendix A:

Assuming that the resonant aperture is located at z=0, and its normal direction is parallel to the z-axis, as sketched in Fig. 1. In light of the Huygens’s theorem, the diffracted field arising from the electrical field at the resonant aperture, $E_t(\rho \text{'},0)$, is express as

(6)$$E(\rho ,z)=2\nabla \times {\displaystyle \int \left[\widehat{z}\times E_t(\rho \text{'},0)\right]G(\rho -\rho \text{'},z)d\rho \text{'}},$$

herein,$\rho ={[xy\text{}\text{}\text{}]}^T,d\rho \text{'}=dx\text{'}dy\text{'},G(\rho -\rho \text{'},z)$is the three dimensional scalar Green’s function in free space. Note that above integral is performed over the radiation aperture$\Omega$, which implies that $E_t(\rho \text{'},0)=0$ for $\rho \text{'}\notin \Omega$. Introducing the far-field approximation of$G(\rho -\rho \text{'},z)\approx \frac{e^{i{k}_0{r}_d}}{4\pi r_d}{e}^{-i{k}_r\cdot \rho \text{'}}$ in Eq. (6), we arrive immediately at

(7)$$E(\rho ,z)=2\nabla \times \frac{e^{i{k}_0{r}_d}}{4\pi r_d}{\displaystyle \int \left[\widehat{z}\times E_t(\rho \text{'},0)\right]e^{-i{k}_{r,\rho }\cdot \rho \text{'}}d\rho \text{'}}.$$

With $k_r=k_0(\sin {\theta }_r\cos {\phi }_r,\sin {\theta }_r\sin {\phi }_r,\cos {\theta }_r)$, ${\theta }_r$ and ${\phi }_r$ are the polar angle and azimuth angle, respectively. Using the specific formula for $E_t(\rho \text{'},0)$ as discussion below, and taking $\widehat{z}\times E_t(\rho \text{'},0)=C{T}^{TM}{e}^{-i{k}_{inc,\perp }\cdot \rho \text{'}}$ into account, we can simplify Eq. (7) into

(8)$$\begin{array}{l}E(\rho ,z)=2\nabla \times \left(\frac{e^{i{k}_0{r}_d}}{4\pi r_d}C{T}^{TM}{\displaystyle \int e^{i\left(k_{inc,\rho }-k_{r,\rho }\right)\cdot \rho \text{'}}d\rho \text{'}}\right)\\ =2L_x{L}_y{T}^{TE,TM}\sin c\left(\frac{L_x\left(k_{r,x}-k_{in,x}\right)}{2\pi }\right)\sin c\left(\frac{L_y\left(k_{r,y}-k_{in,y}\right)}{2\pi }\right)\nabla \times \left(C\frac{e^{i{k}_0{r}_d}}{4\pi r_d}\right)\end{array}$$

where$C$ is a constant vector,$T^{TM}$ are dependent on the transmission coefficients associated with specific TM illumination polarizations.For this TM illumination, the incident plane wave is

(9)$$E_{inc}=(\widehat{x}\cos {\theta }_i\cos {\phi }_i+\widehat{y}\cos {\theta }_i\sin {\phi }_i-\widehat{z}\sin {\theta }_i)e^{i{k}_{inc,z}z}{e}^{i{k}_{inc,\rho }\cdot \rho }.$$

Then, the transversal field at the output of the resonant aperture is

(10)$$E_t=\{\begin{array}{c}\frac{k_{inc,z}}{k_{in,\rho }^2}\sin {\theta }_i{T}^{TM}{e}^{i{k}_{in,z}z}{k}_{in,\rho }{e}^{i{k}_{in,\rho }\cdot \rho },\rho \in \Omega \\ 0,\rho \notin \Omega \end{array},$$

where$T^{TM}=C_p{e}^{i{k}_{pz}d}+D_p{e}^{-i{k}_{pz}d}$, $D_p=-\frac{R_{op}^{TM}{T}_{op}^{TM}{e}^{i2k_{pz}d}}{1-{\left(R_{op}^{TM}\right)}^2{e}^{i2k_{pz}d}}$

$$C_p=\frac{T_{op}^{TM}}{1-{\left(R_{op}^{TM}\right)}^2{e}^{i2k_{pz}d}},R_{op}^{TM}=\frac{k_{in,z}{\epsilon }_r-k_{pz}}{k_{in,z}{\epsilon }_r+k_{pz}},T_{op}^{TM}=\frac{2k_{in,z}{\epsilon }_r}{k_{in,z}{\epsilon }_r+k_{pz}}$$

Appendix B

This appendix provides briefly the iteratively reweighed algorithm (IRA) for solving the convex optimization problem of Eq. (5) [21]. For numerical implementation, Eq. (5) is formally put into:

(11)$$\widehat{O}=\arg {\min }_O\left[{\Vert E-AO\Vert }_2^2+\gamma {\Vert O\Vert }_1\right],$$

herein,$E$ is the 2500-length vector corresponding to 2500 sampling points of frequency-dependent measurement$E\left(\omega \right)$. In our simulations, the region of interest (ROI) where the imaged object is fallen into has been divided into $50\times 50$ subgrids with size of$0.02{\lambda }_0\times 0.02{\lambda }_0$. Correspondingly,$O$ is an unknown vector with size of$2500\times 1$.$A$ is the mapping matrix with size of 2500 by 2500, whose entries are formed by calculating$G(r_d,r\text{'};\omega )E_{in}(r\text{'};\omega )$. The IRA, a popular technique widely used in the community of sparse signal reconstruction is applied to solve Eq. (11), as demonstrated in Table 1. In Table 1,$A\text{'}$ denotes conjugate transpose of$A$,$O(n)$is the nth element of$O$, $\delta$ ($\delta ={10}^{-4}$is used in this paper) is a small positive constant to avoid the singularity for$|O(r\text{'}){|}^2$being zero or almost zero. It is not straightforward that choosing suitable values for the artificial parameter$\gamma$, although some time-consuming methods, such as L-curve and generalized cross validation methods, have been proposed. Regarding the stop criterion, this paper specifies heuristically the maximum iteration step to be 10.

Table 1. The Procedure of IRA Algorithm

View Table

Acknowledgment

This work is supported by National Natural Science Foundation of China (61471006).

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