Feasibility of resonant metalens for the subwavelength imaging using a single sensor in the far field

Lianlin Li; Fang Li; Tie Jun Cui

doi:10.1364/OE.22.018688

1. Introduction

A century ago, Lord Rayleigh formulated that the minimum resolvable size for an imaging system was in the order of the wavelength regardless of the physical apparatus employed for measurements [1]. The resolution limit, referred to as the Rayleigh limit nowadays, is related to a fundamental fact that the information associated with the subwavelength structures of a sample, which is encoded in the evanescent components of the fields emerged from the sample, is exponentially lost in the far-field region. Amongst numerous proposals of surpassing the diffraction limit, the near-field scanning approach (for instance, the near-field scanning optical microscopy (NSOM)) has become an established discipline [2]. NSOM and its variants rely on the use, either direct or indirect, of the evanescent waves, and thus suffer from a practical challenge by requiring at least one of the probing sensors to be within one wavelength distance from the sample surface. Here, we would remark that although in the NSOM the detector (PMT or photodiode) is placed in the far field, the scanning tip should be moved mechanically in the near field of probed objects. The progress made in the near-field microscopy naturally raises a question: i.e., is it possible that the super resolution can be achieved when the probing sensors of collecting data without the requirement of mechanical movements are placed in the far-field region?

The theory of superoscillatory indicates that over a finite interval, a waveform oscillates arbitrarily faster than its highest component in its operational spectrum, and thus makes it possible to encode fine details of the probed objects into the field of view. In light of this theory, several optical devices have been built up to achieve super-resolution imaging from far-field measurements [e.g., 3]. Although the superoscillatory allows us to circumvent the near-field scanning, the obtainable enhancement in resolution is determined by the signal-noise ratio (SNR) to some extent, and requires a huge-size mask with enough fabrication finesse.

More recently, Lemoult and associates made efforts towards the direction of super-resolution imaging from the far-field measurements. They introduced the concept of metalens made of an array of resonators, which supports a collection of the eigenmodes [4–8]. These modes have their own resonant frequencies and distinct far-field radiation patterns as the feature of resonant frequencies. Such conversion of spatial and temporal degrees holds the promising of subwavelength imaging from far-field measurements using the polychromatic light. Although the resonant metalens is capable of producing super-resolution imaging from far-field measurements, the imaging speed is limited by the need for multiple measurements [9].

This paper investigates the feasibility of a resonant metalens for the subwavelength imaging of the probed objects in its near field using a single sensor in the far field. We show that such resonant metalens is capable of converting the evanescent waves emerged from the probed objects into the propagating waves, and is related to the so-called super-resonance. Furthermore, we demonstrate that such super-resonance is capable of producing a super-resolution image of the probed objects when only a single sensor is available. Thus, the super-resonance has its intrinsic role of enhancing the information capacity of an imaging system, and thus leads to the super-resolution imaging from far-field measurements, which is validated by full-wave numerical simulations.

2. Methodology

To illustrate the fundamental principle of the resonant metalens for far-field imaging with subwavelength resolution, we study a two-dimensional (2D) and scalar problem, which can be generalized into three-dimensional full-vector case in a straightforward manner. For the sake of simplicity, the resonant metalens is assumed to consist of a lattice of scatterers (referred to Fig. 1), and these scatterers are characterized by their frequency-dependent electric polarizability $α (ω) = \frac{- 4 Γ c_{0}^{2}}{ω_{p} (ω^{2} - ω_{p}^{2} - i Γ ω^{2} / ω_{p})}$ , as adopted in [10, 13], where $Γ = 10^{11} s^{- 1}$ is the linewidth, $ω_{p} = 3.14 \times 10^{12} s^{- 1}$ is the plasma frequency, and $c_{0}$ is the light speed in vacuum.

Fig. 1 The sketch map for illustrating the principle of the resonant metalens for the subwavelength imaging from far-field measurements. In this figure, the metalens is made of a lattice of 3 × 10 metallic cylinders characterized by Eq. (1). The distance between two neighboring scatterers is d.

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We start the discussion from the Green’s function which plays a fundamental role in analyzing the imaging resolution of the system. As sketched in Fig. 1, the Green’s function $G (r_{d}, r_{s}; ω)$ establishes connection from a probed source located at $r_{s}$ to the field at $r_{d}$ , which is determined by the following coupled-dipole equations [10–12]:

G (r_{d}, r_{s}; ω) = G_{0} (r_{d}, r_{s}; ω) + k^{2} α (ω) \sum_{n = 1}^{N} G_{0} (r_{d}, r_{n}; ω) g_{n}

a n d g_{m} = G_{0} (r_{m}, r_{s}; ω) + k^{2} α (ω) \sum_{n = 1, n \neq m}^{N} G_{0} (r_{m}, r_{n}; ω) g_{n} (m = 1, 2, \dots, N)

Here, N denotes the number of scatterers,

G_{0} (r_{m}, r_{n}; ω) = \frac{i}{4} H_{0}^{(1)} (k | r_{m} - r_{n} |)

is the Green’s function in free space,

H_{0}^{(1)}

is the zero-order Hankel function of the first kind, and

k = ω / c_{0}

is the free-space wave number. Then the compact-form solution to Eqs. (1) and (2) reads:

\begin{array}{l} G (r_{d}, r_{s}; ω) = G_{0} (r_{d}, r_{s}; ω) + k^{2} α (ω) G_{0}^{lens \to far} (r_{d}; ω) \times \\ {(I - k^{2} α (ω) R g G_{0}^{lens \to lens} (ω))}^{- 1} G_{0}^{source \to lens} (r_{s}; ω) \end{array}

in which

G_{0}^{lens \to far} (r_{d}; ω)

is an N-length row vector formed by evaluating the function

G_{0} (r_{d}, r_{n}; ω)

for varying locations

r_{n}

(n = 1,2,…,N),

G_{0}^{source \to lens} (r_{s}; ω)

is an N-length column vector by calculating

G_{0} (r_{m}, r_{s}; ω)

for varying locations

r_{m}

(m = 1,2,…,N), and

R g G_{0}^{lens \to lens} (ω)

is an

N \times N

matrix, whose entries are from

G_{0} (r_{m}, r_{n}; ω)

but has all diagonal elements being zero.

2.1 The degree of freedom (DoF) of a single-sensor imaging system

In this subsection, we demonstrate that the DoF of far-field measurements can be efficiently driven up by using the resonant metalens, in combination with a broadband illumination, since the temporal measurements in the far field not only depend on the lens size, but also the physical property of the lens.

It is demonstrated from Eq. (3) that the DoF of $G (r_{d}, r_{s}; ω)$ with respect to $r_{d}$ for a monochromatic illumination is mostly restricted by $G_{0}^{lens \to far} (r_{d}; ω)$ with the feature of a naturally spatial low-pass filter [14, 15]. Actually, the DoF in relation to $G (r_{d}, r_{s}; ω)$ is merely determined by the sizes of scanning aperture and lens, regardless of the lens physical property [14]. As a result, for a scanning aperture of a given size, the degree of improvement on imaging resolution by the use of the lens is limited by the lens size: the bigger the lens size is, the higher the resolution is. Furthermore, the information capacity of a monochromatic imaging system is of the order $O (T \times B)$ , where B is the spatial bandwidth and T is the size of scanning aperture [16]. Usually, B is fixed for a given T and operational frequency $ω$ . Consequently, the use of any lens (either conventional or man-made) is limited in improving physically the resolution of a monochromatic imaging system in the far-field region, which is common knowledge.

However, above discussions will be considerably changed when a broadband illumination is used. We take the analysis of sensitivity of $G (r_{d}, r_{s}; ω)$ with respect to $ω$ and $r_{s}$ . Taking the gradients of both sides of Eq. (2) with respect to $ω$ leads to

\begin{array}{l} \frac{d}{d ω} g_{m} = \frac{d}{d ω} G_{0} (r_{m}, r_{s}; ω) + \frac{d k^{2} α (ω)}{d ω} \sum_{n = 1, n \neq m}^{N} G_{0} (r_{m}, r_{n}; ω) g_{n} \\ + k^{2} α (ω) \sum_{n = 1, n \neq m}^{N} \frac{d G_{0} (r_{m}, r_{n}; ω)}{d ω} g_{n} \\ + k^{2} α (ω) \sum_{n = 1, n \neq m}^{N} G_{0} (r_{m}, r_{n}; ω) \frac{d g_{n}}{d ω} \end{array}

As

ω

is around the resonant frequency

ω_{p}

, the first and third terms of the right-hand side in Eq. (4) are usually negligible and can be ignored. Then we have

\frac{d}{d ω} g_{m} = \frac{d k^{2} α (ω)}{d ω} \sum_{n = 1, n \neq m}^{N} G_{0} (r_{m}, r_{n}; ω) g_{n} + k^{2} α (ω) \sum_{n = 1, n \neq m}^{N} G_{0} (r_{m}, r_{n}; ω) \frac{d g_{n}}{d ω}

The compact-form solution to Eq. (5) can be immediately derived as

\begin{array}{l} \frac{d G (r_{s}; ω)}{d ω} = \frac{d k^{2} α (ω)}{d ω} {(I - k^{2} α (ω) R g G_{0}^{lens \to lens} (ω))}^{- 1} R g G_{0}^{lens \to lens} (ω) G (r_{s}; ω) \\ = \frac{d k^{2} α (ω)}{d ω} {(I - k^{2} α (ω) R g G_{0}^{lens \to lens} (ω))}^{- 1} (G (r_{s}; ω) - G_{0}^{source \to lens} (r_{s}; ω)) \end{array}

where

G (r_{s}; ω)

is an N-length column vector formed by

{g_{n}}

(n = 1,2,…,N), and the argument

r_{s}

is included to highlight its dependence on the source location . It is noted that for the resonant metalens, the matrix

I - k^{2} α (ω) R g G_{0}^{lens \to lens} (ω)

is usually ill-posed when

ω

is around the plasma frequency

ω_{p}

. Here, we would remark that this ill-posedness is related to a more general concept, that is, the super-resonance phenomenon introduced by Tolstoy in 1986 [17]. The super-resonance was a true resonance of an acoustic system; in the sense that in the absence of radiation damping the system would have anomalously infinite amplitude when driven at the resonant frequencies [18, 19]. For this reason, we would like to use the super-resonance lens instead of the resonant metalens below.

To investigate the capability of the super-resonance lens in resolving two points located at $r_{s 1}$ and $r_{s 2}$ , subject to $| r_{s 1} - r_{s 2} | \leq λ / 8 ~ λ / 10$ , we introduce two notations as

Δ G (ω) = G (r_{s 1}; ω) - G (r_{s 2}; ω),

Δ G_{0}^{source \to lens} (ω) = G_{0}^{source \to lens} (r_{s 1}; ω) - G_{0}^{source \to lens} (r_{s 2}; ω)

Then one can deduce from Eq. (6) that

\frac{d}{d ω} Δ G (ω) = \frac{d k^{2} α (ω)}{d ω} {(I - k^{2} α (ω) R g G_{0}^{lens \to lens} (ω))}^{- 1} (Δ G (ω) - Δ G_{0}^{source \to lens} (ω))

It should be highlighted that both

Δ G (ω)

and

Δ G_{0}^{source \to lens} (ω)

capture the difference of the two fields inside the super-resonance lens emerged from the two sources located at

r_{s 1}

and

r_{s 2}

. When the frequency is in the range of the super resonances of the lens,

| \frac{d k^{2} α (ω)}{d ω} |

becomes extremely large. More importantly, in this range

I - k^{2} α (ω) R g G_{0}^{lens \to lens} (ω)

is strongly ill-posed, leading to the extremely large entries in its reverse. Thus, there are two factors to amplify

Δ G (ω)

and

Δ G_{0}^{source \to lens} (ω)

:

{(I - k^{2} α (ω) R g G_{0}^{lens \to lens} (ω))}^{- 1}

and

\frac{d k^{2} α (ω)}{d ω}

. On the other hand, to ensure both

Δ G (ω)

and

Δ G_{0}^{source \to lens} (ω)

are non-zero, both

r_{s 1}

and

r_{s 2}

should be in the vicinity of the super-resonance lens. Due to the use of the super-resonance lens, we observe that

Δ G (ω)

is remarkably larger than

Δ G_{0}^{source \to lens} (ω)

. Hence we conclude that

Δ G (ω)

is more sensitive to the working frequencies in the super resonance range of the lens, leading to the capability of resolving two objects separated in a subwavelength distance with a broadband illumination.

From the sampling standpoint, such high sensitivity to working frequencies implies that the sampling space on frequency should be sufficiently small, and thus is related to the increase of the independent measurements in the far-field region. Actually, the information capacity of a broadband imaging system is of the order $O (T \times B)$ ; herein B is the temporal bandwidth and T is the duration of the time-domain response. The phenomenon of super-resonance is highly related to the extremely rich multi-scattering effects, which exhibits strong extension on the duration of the temporal response. In this way, the information capacity is increased. Furthermore, we emphasize that the near-field coupling between the imaged objects and super-resonance lens encodes subwavelength details of the objects into temporal measurements in the far-field region. Hence, in order to obtain the subwavelength imaging using a single far-field sensor, we conclude that three basic requirements should be satisfied: the super resonance, the near-field coupling, and the temporal (or broadband) illumination.

2.2 The reconstruction algorithm

Basically, the frequency-dependent measurement data $E (ω)$ acquired at $r_{d}$ , is related to the probed object characterized by its contrast $O (r')$ by $E (ω) = \int_{D}^{} G (r_{d}, r^{'}; ω) E_{in} (r^{'}; ω) O (r') d r^{'}$ , where $E_{in}$ is the incident field on the resonant aperture, and the integral is performed over the region of interest. Note that both $G (r_{d}, r^{'}; ω)$ and $E_{in} (r^{'}; ω)$ are known to be a prior. Regarding the inverse procedure, the object can be retrieved from the frequency-dependent measurements $E (ω)$ by optimizing the following regularized problem, i.e.

\hat{O} (r) = {argmin}_{O (r')} [{| E (ω) - \int_{D} G (r_{d}, r'; ω) E_{in} (r'; ω) O (r') d r^{'} |}^{2} d ω + γ \int_{D} | O (r') | d r']

In Eq. (8), the second term involved in the optimized object function stands for the penalty term to stabilize the optimization procedure, and

γ

is a balance factor. The solution to Eq. (8) can be achieved by using the iteratively reweighed approach [20, 21].

3. Simulation results

In this section, we verify the proposed subwavelength far-field imaging with the use of the super-resonance lens along with a single sensor, as sketched in Fig. 1. Here, the simulation parameters are set as:d = 5.8 μm (around 0.01λ, λ is the central wavelength of the illumination pulse), $x_{0}$ = 11.7μm, N = $20 \times 20$ , $r_{d}$ = (6mm,6mm), and the operational wavelength varies from 578μm to 585μm with a step of 0.002μm.

First, we will investigate the super-resonance property. As discussed previously, the far-field response is highly sensitive to the working frequencies when the super-resonance lens is used. Thus, it is expected that the frequency-dependent measurement acquired at r_d exhibits a large amount of peaks within a certain frequency range when a broadband point source is set in the vicinity of the super-resonance lens. To show this clearly, we excite the super-resonance lens with a z-polarized line source centered at (−11.7, 0)µm, where this source emits a pulse with the frequency-dependent amplitude being unity for the operational wavelength ranging from 578µm to 585µm while being zero otherwise. The frequency-dependent response acquired at $r_{d}$ is provided in Fig. 2(a), and its corresponding time-dependent response, as shown in Fig. 2(b), can be calculated by performing the Fourier transform. It is noted that the time-domain response is complex-valued instead of being real since the response in the frequency domain is not conjugate symmetric. Therefore, only the real part of the response in the time domain is plotted in Fig. 2(b). Additionally, the excitation pulse in the time domain is plotted in the inset of Fig. 2(b). This set of figures demonstrates that a broadband pulse will show rich peaks in the frequency domain after experiencing the super-resonance lens; 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 DoF of the measurements will be considerably driven up with a factor of 100 and beyond.

Fig. 2 The normalized amplitude of the frequency-dependent (a) and time-dependent (b) responses acquired at $r_{d}$ . Here, a point source (as shown in the inset in Fig. 2(b)) is centered at (−11.7, 0) µm. This set of figures shows many abrupt changes within a very small frequency separation, implying that the response is highly sensitivity to frequency. The results are generated by applying the full-wave solver to the Maxwell’s equations, i.e., the coupled dipole method.

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The super-resonance lens is spatial-temporal dispersive, which is capable of encoding the spatial details of the object under consideration in the subwavelength scale into the time domain [6–8]. It is expected that parts of the evanescent waves emerged from the objects under consideration can be converted into the propagating waves after experiencing the super-resonance lens. To show this point clearly, we perform another set of simulations. We assume that the super-resonance lens is illuminated by a plane wave characterized by $E_{i n c} = \hat{z} e^{i (y k_{i n, y} + x k_{i n, x})}$ ,where $k_{i n, x} = \sqrt{k_{0}^{2} - k_{i n, y}^{2}}$ and $k_{0}$ denotes the wave number in vacuum. Note that the case $| k_{i n, y} | > k_{0}$ corresponds to the illumination of an evanescent wave, in contrast to $| k_{i n, y} | < k_{0}$ for a propagating wave. Figure 3 shows the dependence of the electric field scattered from the super-resonance lens acquired at $r_{d}$ on $k_{i n, y} / k_{p}$ ( $k_{p} = 2 π / ω_{p}$ ) and $ω / ω_{p}$ . From this set of figures, we can see that the illumination of the evanescent waves can be efficiently captured by the sensor at the far field, implying the evanescent waves have been converted into propagating waves after experiencing the resonant aperture, as illustrated by the bright region in this figure. Note that the amplitude of the field scattered from the super-resonance lens is comparable to the illumination field, as demonstrated in Fig. 3.

Fig. 3 The dependence of the amplitude of electrical field scattered from the super-resonance lens on $k_{i n, y} / k_{p}$ and $ω / ω_{p}$ . The electrical field is acquired at $r_{d}$ . In this figure, the x-axis is $k_{i n, y} / k_{p}$ and the y-axis denotes $ω / ω_{p}$ . This set of results is generated by applying the full-wave solver to the Maxwell’s equations, i.e., the coupled dipole method.

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Now, we turn to the reconstruction of the probed objects. The probed object consists of dielectric cylinders represented by their polarizabilities, which are applicable to dielectric cylinders with a diameter of 4 µm and relative dielectric constant ranging from 1.2 to 6.4. The distance between the centers of two neighboring objects is set to be 12 µm. We voluntarily opt for a low-refractive-index-contrast object, which is typically a soft matter. In addition, $γ$ is set to be $10^{6}$ in our reconstruction procedure. Figure 4(a) provides the amplitude of the electric fields collected at $r_{d}$ across the range of the operational wavelength from 578 µm to 585 µm, which shows strongly oscillation related to the super-resonance phenomena, as shown by the curve of the condition number. We perform a numerical proof-of-concept investigation, and the corresponding reconstructed results are shown in Fig. 4(b), where the additive Gaussian noise with 40dB SNR has been added to the simulated data. In addition, the simulation data input to the reconstruction procedure is generated by applying a full-wave solver based on the coupled dipole method. To underline the importance of the super-resonance lens, we perform the reconstruction when 60 receivers are equally distributed on a circle with a radius of 6mm, and the reconstruction results are illustrated by the blue line in Fig. 4(b). From Fig. 4, we clearly notice that the super-resonance lens is indeed beneficial to the subwavelength imaging by providing a resolution of λ/20 or even beyond from a single sensor at the far-field region. We also notice the necessity of the three basic requirements for far-field imaging beyond the diffraction limit. We remark that our approach is at the frontier of what is achievable with the current experimental techniques.

Fig. 4 (a) The black line corresponds to the normalized amplitude of electric fields measured by the sensor located at $r_{d}$ as a function of operational wavelength. The red line corresponds to the logarithm of the condition number of $B = I - k^{2} α (ω) R g G_{0}^{lens \to lens} (ω)$ . (b) The reconstruction results using the super-resonance (SR) lens in combination with a broadband illumination with the operational wavelength ranging from 578 µm to 588 µm by a step of 0.002 µm, denoted by the red line. For comparison, the ground truth (black line) and the results without the super-resonance lens (blue line) are also provided. (The reader of interest can get the Matlab Code for reproducing above results by sending a request email to lianlin.li@pku.edu.cn)

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Finally, we will examine the effects of both the distance of the detector and the SNR level on the reconstruction quality. Figure 5 compares the reconstruction results for different distances of the acquiring sensor, where the rest of simulation parameters are the same as those used in Fig. 4. From this set of curves, one can see that with the considered resonant metalens the reconstruction quality is not very sensitive to the working distance of the single sensor overall, and that it will be decreased more or less with the growth of the distance of the sensor. The possible reason is as follows. The whole propagating waves acquired by the sensor in the far field consist mainly of two parts: one is from the original propagating illumination illuminated onto the metalens, and the other is converted from the evanescent waves. However, the latter is weaker in contrast to the former more or less, as illuminated in Fig. 3, especially for large distance since the latter are emerged from the interaction between the resonant metalens and the evanescent waves illuminated on it. More in-depth physical investigation about it should be carried out in the near future.

Fig. 5 The reconstruction results using the super-resonance (SR) lens for four different working distances of sensor, where the simulation parameters are the same as those used in Fig. 4. The ground truth is also provided.

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Figure 6 presents the comparison of the reconstructions achieved by the proposed imaging methodology for different SNR levels, where the rest of simulation parameters are the same as those in Fig. 4. We have added white Gaussian noise (WGN) with signal-to-noise ratio values of 40dB, 35dB, 30dB, 25dB and 20 dB to the simulated field acquired by a single sensor. It is observed that the higher the SNR is, the better the reconstruction quality is, and that the proposed imaging system of a single sensor is robust to the noise up to the SNR being 30dB.

Fig. 6 The reconstruction results using the super-resonance (SR) lens for different noise levels of 40dB, 35dB, 30dB, 25dB and 20dB, where the simulation parameters are the same as those used in Fig. 4. The ground truth is also provided.

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4. Summary

In summary, this paper investigates the feasibility of the resonant metalens for the far-field imaging beyond the diffraction limit using a single sensor. It is shown that the resonant metalens is related to the super-resonance phenomenon. We show that for a given finite operational bandwidth, the super-resonance lens can efficiently increase the information capacity of the measurements in the far field, leading to the far-field imaging beyond the Rayleigh limit. Furthermore, we demonstrated that three basic requirements should be satisfied to obtain subwavelength imaging from far-field measurements: the super-resonance, the near-field coupling, and the broadband illumination. The performance of a single sensor imaging system is verified by full-wave simulations. Unlike most current imaging hardware, this system gives access to full compressive measurements by a single antenna, drastically speeding up the acquisition. Such implementation can find applications in other disciplines, such as microwave, optics and ultrasound imaging.

References and links

1. L. Rayleigh,“On pin-hole photography,” The London, Edinburg and Dublin philosophical magazine and journal of science, 5, 31 (1891).

2. http://en.wikipedia.org/wiki/Near-field_scanning_optical_microscope

3. E. T. F. Rogers and N. I. Zheludev, “Optical super-oscillations: sub-wavelength light focusing and super-resolution imaging,” J. Opt. 15(9), 094008 (2013). [CrossRef]

4. F. Lemoult, J. de Rosny, M. Fink, and G. Lerosey, “Resonant metalenses for breaking the diffraction barrier,” Phys. Rev. Lett. 104(20), 203901 (2010). [CrossRef] [PubMed]

5. F. Lemoult, M. Fink, and G. Lerosey, “Far-field sub-wavelength imaging and focusing using a wire medium based resonant metalens,” Waves in Random and Complex Media 21(4), 614–627 (2011). [CrossRef]

6. F. Lemoult, M. Fink, and G. Lerosey, “Revisiting the wire medium: an ideal resonant metalens,” Waves in Random and Complex Media 21(4), 591–613 (2011). [CrossRef]

7. F. Lemoult, M. Fink, and G. Lerosey, “Dispersion in media containing resonant inclusions: where does it come from,” 2012 Conference on, Lasers and Electro-Optics (2012). [CrossRef]

8. F. Lemoult, M. Fink, and G. Lerosey, “A polychromatic approach to far-field superlensing at visible wavelengths,” Nat. Commun. 3, 1885 (2012).

9. D. Lu and Z. Liu, “Hyperlenses and metalenses for far-field super-resolution imaging,” Nat. Commun. 3, 1205 (2012).

10. R. Pierrat, C. Vandenbem, M. Fink, and R. Carminat, “Subwavelength focusing inside an open disordered medium by time reversal at a single point antenna,” Phys. Rev. A. 87, 041801 (2013).

11. P. C. Chaumet, A. Sentenac, and A. Rahmani, “Coupled dipole method for scatterers with large permittivity,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(3), 036606 (2004). [CrossRef] [PubMed]

12. P. C. Chaumet, K. Belkebir, and A. Rahmani, “Coupled-dipole method in time domain,” Opt. Express 16(25), 20157–20165 (2008). [CrossRef] [PubMed]

13. P. de Vries, D. V. van Coevorden, and A. Lagendijk, “Point scatterers for classical waves,” Rev. Mod. Phys. 70(2), 447–466 (1998). [CrossRef]

14. F. Simonetti, M. Fleming, and E. A. Marengo, “Illustration of the role of multiple scattering in subwavelength imaging from far-field measurements,” J. Opt. Soc. Am. A 25(2), 292–303 (2008). [CrossRef] [PubMed]

15. O. M. Bucci and G. Franceschetti, “On the degrees of freedom of scattered fields,” IEEE Trans. Antenn. Propag. 37(7), 918–926 (1989). [CrossRef]

16. I. J. Cox and C. R. Sheppard, “Information capacity and resolution in an optical system,” J. Opt. Soc. Am. A 3(8), 1152–1158 (1986). [CrossRef]

17. I. Tolstoy, “Superresonant systems of scatters I,” J. Acoust. Soc. Am. 80(1), 282–294 (1986). [CrossRef]

18. G. S. Sammelmann and R. H. Hackman, “Acoustic scattering in a homogeneous waveguide,” J. Acoust. Soc. Am. 82(1), 324–336 (1987). [CrossRef]

19. The super-resonance is mathematically that the matrix $B = I - k^{2} α (ω) R g G_{0}^{lens \to lens} (ω)$ in Eq. (3) is strongly ill-posed, which means that the ratio $\frac{σ_{1}}{σ_{N}}$ (i.e., the condition number) is very large, where $σ_{1}$ is the first singular value (the maximum) of the matrix, and $σ_{N}$ is the final (the minimum) singular value.

20. L. Li and B. Jafarpour, “Effective solution of nonlinear subsurface flow inverse problems in sparse bases,” Inverse Probl. 26(10), 105016 (2010). [CrossRef]

21. M. Elad, Sparse and Redundant Representations: From Theory to Applications in Signal and Image Processing (Springer Press 2010).

Feasibility of resonant metalens for the subwavelength imaging using a single sensor in the far field

Abstract

1. Introduction

2. Methodology

2.1 The degree of freedom (DoF) of a single-sensor imaging system

2.2 The reconstruction algorithm

3. Simulation results

4. Summary

References and links

Cited By

Figures (6)

Equations (10)

Optics Express