1 Introduction

Negative-refraction materials, as a new kind of artificial electromagnetic media, have attracted great interest in recent years. Negative-refraction materials with both negative refractivity and negative permeability are usually known as left-handed materials (LHM), since the electric field vector, the magnetic field vector and the wave vector of electromagnetic wave form a left-handed triplet in LHM. In the year of 1968, Veselago [1] investigated the anomalous phenomena of electromagnetic wave propagating in LHM theoretically, and then in 2000, Smith et al. [2] demonstrated firstly such LHM in microwave band in their laboratory. Based on abnormal electromagnetic properties, LHM have found many important implications in optical and microwave areas, such as in the area of near-field target detection and imaging [3, 4]. Among them, the so-called perfect lens made of LHM with no losses is more attractive because of its high focus resolution overcoming the optical diffraction limit [5]. In general, the higher focus resolution yields the better imaging resolution in near-field target detection and imaging.

Though, in theory, a flat slab of LHM could be used as a ‘perfect lens’ for near focusing beyond the diffraction limit, there is still much uncertainty about whether such materials actually exist in nature. Using different methods, Notomi et al. [6] have demonstrated 2D photonic crystals exhibiting negative index or negative refraction effects, namely negative-refraction photonic crystals (NR-PC). From their results, we know that the negative refraction usually occurs in the band gap of the equifrequency surface (EFS) in k space, because the size of the contour of EFS in the k space decreases with increasing frequencies. And the contour of EFS up to a certain frequency takes shape of a quasi-circular [7], which means the light propagating in PC is similar to that propagating in the isotropic medium at these frequencies. Thereby, effective negative index of refraction (n eff) can be used to describe the propagation property of light with frequency falling into a certain frequency spectrum for a given NR-PC, and the NR-PC could be used as a flat LHM lens in the near-field target detection and imaging [8, 9].

Basing on the phenomena of negative refraction, it is easy to vision that the NR-PC flat lens and the LHM flat lens share the same image-forming principle, as shown in Fig. 1a, b. Lightwave emitted from a point source on one side of the NR-PC/LHM lens can be focused at one focal point F 1 inside the NR-PC/LHM lens firstly, and then be focused at the other focal point F 2 outside the NR-PC/LHM lens. When a target is brought into the focal point F 2, lightwave from the source is focused on the target by the flat NR-PC/LHM lens, and the target will backscatter the focused lightwave, then the scattering signal from the target will be refocused in the vicinity of the source point by the same flat NR-PC/LHM lens.

Fig. 1
figure 1

a Target detection and imaging using NR-PC flat lens. b Target detection and imaging using LHM flat lens. c Change curve of n eff and ω in the NR-PC lens (for TM mode)

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The photonic crystal structure examined in this paper is formed by periodically drilling seven rows (in Z direction) of identical air holes with 30 air holes (in X direction) in each row in a GaAs matrix with dielectric constant ε = 12.96 (n = 3.6), as shown in Fig. 1a. And the air cylinders form a triangular array, here the radius of the air cylinders is 0.4a (a represents the lattice constant). The effective refractive index of the photonic crystal is calculated with TM mode and is drawn in Fig. 1c using the algorithm in Ref. [6]. It is understandable that the n eff changes with normalized frequency ω = a/λ. From Fig. 1c, we know that, when n eff takes the value of −1, the corresponding normalized frequency ω is about 0.3068.

Focus-scanning scheme has been proved to be effective for the detection and imaging of targets. And with the NR-PC flat lens defined in Fig. 1a, the proposed focus-scanning scheme in the paper is essentially carried out by moving the source–detector pair together laterally (X directionally), scanning the target (the focal point F 2) in the region under detection, measuring the target’s scattering signal refocused by the flat NR-PC lens, and finally, screening the distribution of the scattering field level.

Here, we remark that when a target is put on the focal point F 2, the field level recorded by the moving detector is actually the total lightwave field, which is the compound of three parts: the wave emitted from the source, the wave reflected from the entrance and exit surfaces of NR-PC lens (for lens of n eff ≠ −1), and the scattering signal backscattered from the target. Therefore, the scattering signal is achievable by taking the difference between the values recorded by the detector with/without a target on the focal point F 2.

Our study is based on the works [10, 11]. They demonstrated the influence on light focusing of defect, while we mainly study the effect on target detection and imaging. Numerical simulations with the finite-difference time-domain (FDTD) method show that a sharp transmission peak of the lightwave appears at the resonance frequency 0.3068(a/λ) for the NR-PC flat lens with cylinder air holes. In addition, the lightwave backscattered from the target is enhanced greatly, which significantly improves the lateral refocusing and imaging resolution, and as a result, optimizes the performance of the target detection and imaging system. Detailed comparison between the scattering signals obtained with and without use of the NR-PC lens will be more helpful to evaluate flat NR-PC lens’ specific application on target detection and imaging. In addition, we use some defective cylinder air holes NR-PC lens, and further study the effects of defects on target detection and imaging.

2 Design and calculation models of the NR-PC flat lens

The Maxwell’s equations in the photonic crystals can be written as

$$ \frac{{\partial \varvec{H}}}{\partial t} = - \frac{1}{{\mu_{0} \mu (r)}}\nabla \times \varvec{E}, $$
(1)
$$ \frac{{\partial \varvec{E}}}{\partial t} = \frac{1}{{\varepsilon_{0} \varepsilon \left( r \right)}}\nabla \times \varvec{H}, $$
(2)

where \( \nabla = \frac{\partial }{\partial x}\varvec{i} + \frac{\partial }{\partial y}\varvec{j} + \frac{\partial }{\partial z}\varvec{k} \) is the Hamiltonian operator, E and H are the electric and magnetic field vectors of the electromagnetic wave, the permittivity ε(r) is the relative dielectric constant, and μ(r) = 1 is the relative magnetic permeability.

The FDTD method [12] is well established with a somewhat lower computational capacity in comparison to its high accuracy when compared with other methods. It has been widely used to study the characteristics of electromagnetic waves in NR-PC. Its fundamental principle is that the Maxwell’s equations are first expressed as scalar equations of electric and magnetic field components in Cartesian coordinates, and then the differential quotient is replaced with the difference quotient with accuracy to the second order. In our simulation, a perfectly matched layer (PML) is used in the X and Y directions as boundary conditions [12]. Because these equations are the functions of space and time, they can be discretized in the space and time domains by the Yee-cell technique and be used to find field solutions numerically.

It is noteworthy that the mode of light propagating in photonic crystals (PC) is very different from that in LHM. For the LHM with refractive index n = −1, there is no reflection at the air–LHM interface, however, light will experience multiple reflections and refractions at the air–PC interface, even for NR-PC when n eff = −1, leading to great losses or much lower transmissivity for the light propagating through the NR-PC. To optimize the performance of the focus-scanning scheme, an effective way to improve transmissivity is needed.

The corresponding investigation of raising the transmissivity has already been presented in our previous thesis cited by OPTIK [13]. When the center frequency (ω p) of the wave source is set at 0.3068(a/λ), the transmission is enhanced dramatically, with the transmission coefficient up to 4,500 at the frequency point of 0.3068(a/λ). The physical mechanism can be explained by the redistribution of optical energy. The incident optical waves with a different frequency will experience intensive Bragg scattering when they are incident upon the NR-PC and propagating in the NR-PC because of the periodic distribution of negative-refraction media, which results in mini-forbidden bands and a photonic tunneling effect for a given NR-PC [14, 15]. At the same time, the optical energy is highly localized, and the high transmissivity appears at the resonance frequency.

3 Refocusing of the backscattered wave in target detection and imaging from an NR-PC flat lens

The focusing–refocusing property is a very important performance parameter that supports the use of the NR-PC flat lens for lightwave target detection and imaging. By taking the measured full width half maximum (FWHM) for the given energy carried by the focused beam [16] or the width at 0.707-maximum of the normalized field intensity of the refocused beam profile as the definition of resolution, the performance of the target detection and imaging system based on the NR-PC flat lens can be further evaluated. In particular, for the detection and imaging of a small target at an early stage, high sensitivity is very desirable, thus, our investigation may have great significance for imaging systems.

In our FDTD simulations, we consider a typical detecting and imaging system, which consists of a point source–detector pair and a flat NR-PC lens, and the center frequency of the point source is adjusted to 0.3068(a/λ) for a high transmissivity. Meanwhile, the source–detector pair were set to move together along the scanning line z = −λ with intervals of Δx = 0.2 μm. The 2D flat NR-PC lens with a width of d = 2λ is set at 0 ≤ z ≤ 2λ, and a target of a PEC square with a side length of L = 1/3λ is located at the focal point F 2.

First the performance of the complete lightwave is investigated. The simulation diagram of the complete lightwave field in the computation area and its corresponding field intensity distribution along the scanning line z = −λ are depicted in Fig. 2a, b.

Fig. 2
figure 2

a Simulation diagram of the complete lightwave in the computation area. b Lateral beam profile of the complete lightwave field intensity along line z = −λ

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From the arrowheaded lines in Fig. 2a, it is clear that the imaging of the NR-PC lens obeys geometrical optics. Furthermore, there is a one-to-one correspondence between the energy distribution of the complete lightwave field (Fig. 2a) and the distribution curve of its field intensity (Fig. 2b), and the maximum field intensity on the line z = −λ is obtained in the vicinity of the point source.

Figure 2 depicts the properties of the complete lightwave, whereas the main role of NR-PC flat lens in target detection and imaging is actually embodied in the characteristics of the scattering signal from the target. Therefore, to obtain the scattering wave, we may substrate the fields recorded under situation without the target at F 2 from the fields recorded under situation with the target at F 2. Then, on the basis of the other parameters being set the same with the previous preferences, we further contrast the properties of the scattering signal with that obtained when ω p = 0.2068(a/λ); detailed comparison of the property of the scattering signal are shown in Fig. 3.

Fig. 3
figure 3

Beam profile of the normalized field intensity of the scattering signal obtained by detecting the square target of L = 1/3λ using the NR-PC lens when ω p = 0.2068(a/λ) and ω p = 0.3068(a/λ), respectively

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By measuring the full width at 0.707-maximum of the beam profile given in Fig. 3, we find that when using the NR-PC lens for detection, the refocusing resolutions are approximately 0.3718λ and 1.5916λ for ω p = 0.3068(a/λ) and ω p = 0.2068(a/λ), respectively. Obviously, the lateral refocusing resolution corresponding to 0.3068(a/λ) is quadruple higher than that of 0.2068(a/λ).

For further comparison, we also investigate the detection system without using the NR-PC lens. Detailed comparison of the scattering signal obtained with and without the use of the NR-PC lens when ω p = 0.3068(a/λ) is shown in Fig. 4. The results show that due to the use of the NR-PC lens, the refocusing resolution equals approximately to 0.3718λ, which is at least fourfold improvement if compared to 1.6186λ (the refocusing resolution of the directly scattering signal without using the NR-PC lens).

Fig. 4
figure 4

Beam profile of the normalized field intensity of the scattering signal obtained by detecting the square target of L = 1/3λ with and without the NR-PC flat lens when ω p = 0.3068(a/λ)

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From Figs. 3 and 4, we may have the conclusion that there is significant improvement in the scattering signal, which is attributable to the use of the NR-PC lens and the choice of ω p = 0.3068(a/λ). It is intelligible from the standpoint of theory of the mini-forbidden band and resonance excitation effect, as well as the focusing characteristics of the NP-PC lens and the exponential amplification of the evanescent wave [17, 18].

4 The influence of defective NR-PC flat lens on target detection and imaging system

It has been demonstrated that the refocusing resolution will be improved due to the use of NR-PC lens. Further research should be done to find the other factors which will affect the image resolution. The NR-PC flat lens with cylinder holes we use above is all symmetrical and it is perfect. Thus, we take the middle-row cylinder holes with some different defects, such as with one hole moved (Fig. 5a, left), with two discontinuous holes moved (Fig. 5b, left), with three continuous holes moved (Fig. 5c, left), with four discontinuous holes moved (Fig. 5d, left), with half discontinuous holes moved (Fig. 5e, left). Without other changes, by dynamic scanning, we may continue to study the corresponding beam profiles of the normalized field intensity of the scattering signals, as shown in Fig. 5a–e (right) and Table 1. We can introduce defective mode by moving the middle air hole out, or cutting the two discontinuous holes, etc. The defective NR-PC flat lenses we consider are shown in Fig. 5.

Fig. 5
figure 5

Left Structure. Right Normalized field intensity of the refocused beam. The NR-PC flat lens with (a) one hole, (b) two discontinuous holes, (c) three continuous holes, (d) four discontinuous holes, (e) half discontinuous holes moved in the intermediate row

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Table 1 Comparison of the refocusing resolution of the NR-PC slab lens with different defects obtained by detecting a square target of L = 1/3λ
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From Fig. 5a–e and Table 1, we have the observation that the refocusing resolution has been approximately improved by 48, 87, 13, 70, 22 % due to the defects [with one hole moved (Fig. 5a), with two discontinuous holes moved (Fig. 5b), with three continuous holes moved (Fig. 5c), with four discontinuous holes moved (Fig. 5d), with half discontinuous holes moved (Fig. 5e)] by dynamic scanning scheme. We must point out that the other parameters of the lens are kept the same with perfect NR-PC flat lens as shown in Fig. 1a. By comparison, the defective mode of the NR-PC plat lens with two discontinuous holes moved in the intermediate row gives the best refocusing resolution. In summery, the refocusing resolution is improved significantly by introducing the defects when compared with the perfect one.

To further demonstrate the important role of the defective structure (Fig. 5b, left) in target detection and imaging, we choose the structure to do further study as shown in Fig. 6a. Setting the radii of the two holes pointed out in Fig. 6 to 0.1a, 0.2a, 0.3a, respectively, while keeping the other parameters the same with perfect lens, we can get the corresponding beam profiles of the normalized field intensity of the scattering signals based on the structure for target detection and imaging, which is shown in Fig. 6b and Table 2.

Fig. 6
figure 6

Beam profiles of the normalized field intensity of the scattering signal obtained by detecting a square target of L = 1/3λ using NR-PC slab lens with different holes in defects. a The structure diagram of two air holes in defects, b the complete figure, c the partial figure

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Table 2 Comparison of the refocusing resolution of the NR-PC slab lens with different holes in defects obtained by detecting a square target of L = 1/3λ
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From Fig. 6 and Table 2, we have the observation that the refocusing resolution increases more and more quickly while the air-hole radius is set to 0.1a, 0.2a and 0.3a, respectively. There were approximately 46, 21 % decrease and 15 % improvements in refocusing resolution, when compared to that of the structure without the holes (Fig. 5b, left). But the refocusing resolution is improved when compared to the perfect NR-PC flat lens (Fig. 3). And it is exciting that the refocusing resolution of the defective structure with r = 0.3a is improved by 119 % compared to that of the perfect lens.

5 Conclusions

We applied 2D FDTD method and dynamic scanning scheme to study the characteristics of the NR-PC flat lens with defects. It was demonstrated that because of the influence of the mini-forbidden band and resonance excitation effect, high transmissivity will appear at the resonance frequency of 0.3068(a/λ) when the lightwave goes through the NR-PC lens. In addition, the focusing characteristics of the NR-PC lens and the exponential amplification of the evanescent wave [5] make the scheme quite efficient in raising the backscattered wave, which leads to a refocusing resolution and imaging resolution with significant enhancement. More studies demonstrated that the refocusing resolution is improved by introducing defects to the NR-PC flat lens. Furthermore, we can also get better resolution by appropriately decreasing the radius of the defective cylinder (R = 0.3a). In conclusion, our investigation optimizes the performance of the focus-scanning scheme, and provides the basis for converting idealized LHM lens into physically realizable NR-PC flat lens.