A new gradient-index (GRIN) lens that can realize enhanced spatial Fourier transform (FT) over optically long distances is demonstrated. By using an anisotropic GRIN metamaterial with hyperbolic dispersion, evanescent wave in free space can be transformed into propagating wave in the metamaterial and then focused outside due to negative-refraction. Both the results based on the ray tracing and the finite element simulation show that the spatial frequency bandwidth of the spatial FT can be extended to 2.7k0 (k0 is the wave vector in free space). Furthermore, assisted by the enhanced spatial FT, a new long-distance (in the optical far-field region) super-resolution imaging scheme is also proposed and the super resolved capability of λ/5 (λ is the wavelength in free space) is verified. The work may provide technical support for designing new-type high-speed microscopes with long working distances.
© 2015 Optical Society of America
In Fourier optics, spatial Fourier transform (FT) is essential for applications such as performing mathematical operations , spatial frequency filter for optical information processing  and imaging. The most shared devices to realize spatial FT are optical lenses. Even though lenses composed by a single homogeneous material are easy to be fabricated, their limited numerical apertures (NA) result in the limited imaging resolution. Many efforts have been taken to increase the NA by introducing various high index materials or inhomogeneous metamaterials [2, 3], in which immersion techniques [4–6] are the most well-established. However, the resolution improvement by immersion lenses still cannot beat the diffraction limit of λ/2n, λ is the wavelength in free space, n is the index of the immersion material. And the increasing of the NA still remains modest due to the lack of available high index transparent materials. In order to manipulate incident light in the subwavelength scale, NA should be further enlarged to guarantee that evanescent waves with high spatial frequencies can be also focused in the far field. For this purpose, high spatial frequencies must be firstly transformed to low spatial frequencies. Extremely anisotropic materials with indefinite permittivity tensor are well competent for this job [7, 8]. Some other techniques of realizing this transformation are such as nanodot coupling with a surface plasmon polariton condenser  and waveguide modes coupling in metallic arrays . Of course, a lens with a super NA can be directly used for super-resolution imaging [11–18] even in the far filed such as the hyperlens [11, 12, 18].
However, the surfaces of conventional lenses, immersion lenses and hyperlenses are always shaped to be curved for the purpose of phase compensation or momentum compression. This inherent characteristic is not favorable for system integration and practical applications. It may also arouse aberrations in imaging . One can employ different types of metasurfaces [20–24], meta-transmitting arrays  or nano-scale plasmonic waveguide couplers [26, 27] to design plane lenses with flat surfaces, but they again face the same problems discussed above, i.e. the limited NA.
In this paper, we designed and analyzed a new plane gradient-index (GRIN) lens with flat input and output surfaces. With the help of the hyperbolic metamaterials, the NA of the lens can be enlarged up to 2.7 while only low index materials (the index of the material is much smaller than 2.7) need to be used to construct the lens. An effective method based on the ray optics theory is also presented to depict wave propagating and focusing in the hyperbolic metamaterial. Assisted by the enhanced spatial FT of the lens, we further propose a new long-distance (actually work in the far-field region) and high-speed super-resolution imaging scheme, which may find promising applications such as the microscopes with long working distances.
2. Theory of wave propagating in the lens
2.1 Evanescent wave transformation and negative-refraction
As discussed above, in order to realize the enhanced spatial FT with extended input spatial frequency bandwidth (corresponding to a lens with an enlarged NA), evanescent waves in free space must be transformed into propagating wave, otherwise they cannot penetrate into the materials and cannot be projected to the far field, which will lead to the losing of high spatial frequency information. In this way, the spatial frequency bandwidth and the imaging resolution are limited.
Here, we use hyperbolic metamaterials (i.e. the permittivity and permeability are anisotropic and partially negative) to realize this kind of transformation. As we all known, even though a negative refractive-index material  has been successfully constructed by combining the periodic split rings with metal pillar arrays, it is still a challenge to control and adjust the permeability of metamaterials. However, when the transverse magnetic (TM) polarization is only considered, the permittivity plays a key role in controlling the wave propagating .
The dispersion of a two-dimensional (2D) homogeneous isotropic or anisotropic material can be expressed byEq. (1), any evanescent wave in air (kx12<0) can be theoretically transformed into propagating wave (kx22>0) in this kind of hyperbolic metamaterials considering the boundary condition of the transverse wave-vectors must being equivalent at the interface of a two-layer medium. The dispersion curve of the hyperbolic metamaterials is shown in Fig. 1(a).
Unlike the conventional isotropic materials, when electromagnetic wave propagates in hyperbolic metamaterials, the directions of energy flow and phase velocity are not parallel now. For a three layers model (air-metamaterial-air), as illustrate in Fig. 1(b), the incident TM-polarized plane wave is assumed to be with the form Hz1 = exp(–ikx1–iky1), in which kx1 and ky1 are the longitudinal and transverse momentums in the first air layer, respectively. According to the boundary condition and the energy flow (i.e. poynting vector) formula P = E × H, we get
Because εx<0 and εy>0, Eq. (2) can be simplified as Py /Px = –η ky2 /kx2, in which η is a positive number. So when electromagnetic waves propagate in the hyperbolic metamaterial, negative-refraction will occurs at the interface of air-metamaterial, which is also demonstrated in Fig. 1(b). Different from the well-recognized negative-refraction occurring in left-handed metamaterials , this kind of negative-refraction is obtained due to a specific form of isofrequency contours and only emerges for TM-polarized incidence . The relative permittivity of the hyperbolic metamaterial used in Fig. 1(b) is ε = (–1, 1). The amplitude of the incident beam fulfills Gaussian distribution with the beam radius being 2.5λ, defined as the distance between the maximum amplitude and 1/e of the maximum amplitude, the same below.
From the above discussion, we can get two important points: (a) hyperbolic metamaterials are well competent to realize the transformation from evanescent waves to propagating waves; (b) negative-refraction emerges at the interfaces of air-metamaterial. These two points are very crucial for realizing long-distance enhanced spatial FT.
2.2 Wave convergence and momentums compression
Obviously, a single plane homogeneous hyperbolic metamaterial cannot realize focusing due to the lack of phase compensation mechanism. It can be solved by introducing GRIN in hyperbolic metamaterials. In this way, the geometry of a GRIN lens can be designed to be a slab. Different from Ref , we just introduce GRIN in one direction. Even though this will bring additional fabricating difficulty, it is more convenient and flexible for our design. More importantly, it will bring some improvements such as the decreasing of reflections and the reduction of the lens’ thickness, which will be demonstrated below.
The permittivity distribution of the slab fulfills εx = –1, εy = εc (1 + ay2), in which a is a positive constant, εc is the permittivity at y = 0. To analyze waves propagating in this kind of metamaterial, we divide the slab into numerous ultrathin layers, as shown in Fig. 2(a). The thickness of each layer is dy. If dy is much smaller than the height of the slab, each of the layers can be treated to be homogeneous. According to Eqs. (1) and (2), the modified Snell law can be expressed by Eq. (3) can be simplified as tan2(θ2) = εy1/[εy2(εy1–εy2)]. Only when εy1>εy2, θ2 can be positive, which means incident wave will converge in the slab under this condition. For oblique incidence cases, in order to realize wave convergence in the slab, the same conclusion of εy1>εy2 can also be made. That is why a must be a positive constant.
What deserves to be pointed out is that, when incident waves transmit through the slab, not only the wave convergence is realized but also transverse momentums compression is done. To explain it, we use ray optics theory to approximately build the connection between the incident transverse momentum kyin and the output transverse momentum kyout. This approximation is enough accurate for qualitative analysis. After substituting εx = –1 into Eq. (1), we haveFig. 2(b). Due to the transverse momentums conservation at the interface of each layers, kx1 = kx2 = … = kxn, we can deduce thatEq. (5), if εyout >εyin, we have |kyout|<|kyin|, which indicates that the transverse momentum of the incident wave will be compressed after transmitting through the lens. Because the focal positions of evanescent incident waves are vastly deviated from the central axis of the lens, εyout >εyin is credible for paraxial incidence and |kyout|<k0 can be always fulfilled, which will be further demonstrated based on the ray tracing and full-wave simulation below. In this way, the high spatial frequency information can be projected to the far field with the help of the hyperbolic GRIN lens.
2.3 Theoretical verification of double focusing
Based on the above analysis, in this part, we use ray tracing to verify the performance of the lens. The validity of the method has been checked by the model of the conventional isotropic GRIN lens, whose focal positions can be expressed with an analytical form . We find that the simulating results are perfectly matched with the analytical form (not shown here for saving space).
As illustrated in Fig. 2(c), we can recognize the double focusing inside and outside the lens for both the four incidences with the transverse momentums being ky /k0 = 0, 0.5, 1.5, –2, respectively. The thickness and width of the lens are L = 10λ and W = 30λ, respectively. The permittivity of the lens is εx = –1, εy = 2 + 0.069(y/λ)2. The second focusing outside the lens arises from the negative-refraction at the interface of the lens and air. It mainly contributes to the spatial FT over long working distances in air. In fact, even the minimum distance is larger than 2λ, which is superior to those near-sight superlenses with extremely short working distances . What should be also pointed out is that, the cases of ky = 1.5k0 or –2k0 correspond to the evanescent incident waves. It means that the analysis for the momentums compression in part 2.2 is verified. So as we have expected, the input spatial frequency bandwidth of the FT can be theoretically extended and the working distance can be significantly increased by using the lens.
3. Full-wave simulation of the lens
In this part, we use full-wave simulation to further verify the performance of the lens, where the commercial software Comsol Multiphysics is employed. The parameters of the lens are same as in Fig. 2. Here, we only consider the y-direction focusing. For TM-polarization, the electric fields are located in the x-y plane, so the z-direction size of the lens has little influence on the y-direction focusing. Even if the boundary effect arising from the finite z-direction size is considered, its influence to the performance of the lens can also be neglected if only the practical z-direction size of the lens is not too small (for example, smaller than ten wavelengths). So, for theoretical verification, the z-direction length of the lens can be assumed to be infinite. Corresponding to Fig. 2(c), we illustrate the four different incidences with ky /k0 = 0, 0.5, 1.5, –2 in Figs. 3(b)-3(e). As shown in these figures, the double focusing for both cases is also clearly identified, and the focusing patterns are highly consistent with the theoretical results presented in Fig. 2(c). The long profiles of the external focuses make the lens beneficial for the detection. The excitation sources of the simulations are plane waves with the corresponding transverse momentums shown inset each of the figures. The amplitudes of the plane waves fulfill Gaussian distribution with the beam radius being 3λ, which guarantees that the illuminations are in the paraxial region. The excitation sources are λ/5 away from the input interface of the lens, as shown in Fig. 3(a). One can also note that, the y-direction positions of the focuses are all reverse. That is because the directions of the energy flow and phase velocity in the hyperbolic metamaterial are not in a same path, as illustrated in Fig. 1(b) and Fig. 2(a).
Figure 4 presents the y-direction internal and external focal positions of ky /k0 = 0∼2.7, predicted by both the full-wave simulation and the ray tracing. From the figure, we can note that both the absolute values of the y-direction internal and external focal positions approximately increase linearly with the increasing of ky, which fulfills the requirement of the spatial FT. Even though there exist small deviations of the external focal positions, it is inessential for the spatial FT and the imaging. Same as the y-direction focal positions, the x-direction focal positions also increase linearly with the increasing of ky, which is not what we expect. For these reasons, the spatial spectrum must be observed along a non-straight path in air, which can be expressed by x ≈ λ (15.33 −10.9|y|).
4. Super-resolution imaging
4.1 Single-lens super-resolution imaging
The above results reveal that, the NA of the lens can be enlarged up to 2.7. So the spatial frequency bandwidth of the FT can be theoretically extended to 2.7k0. To verify it, we consider a rectangular pulse p(y) as the excitation source and observe the spatial spectrum along the path given above, i.e. x ≈ λ(15.33 −10.9|y|). The width of the pulse is 2λ/3. The theoretical spatial spectrum S(ky) of the source can be expressed byFig. 5(a), the observed spectrum is in a good agreement with the theoretical result except for some small local differences. The magnetic field distribution is also given inset of the Fig. 5(a).
When we use the lens for imaging, the extraordinary NA will help us to beat the diffraction limit. To demonstrate it, we replace the excitation source with two rectangular pulse separated by λ/5. The width of each pulse is λ/15 and the sources are λ/15 away from the input surface of the lens. We also observe the spatial spectrum along the path given inset of Fig. 5(a). The images of the sources can be reconstructed by doing inverse FT (IFT) on the observed spatial spectrum, which is expressed byFig. 5(a). As shown in Fig. 5(b), the two pulses are clearly resolved after the post-processing. In this way, the super resolved capability of λ/5 is realized assisted by the enhanced spatial FT. Because the observation path is optically long-distance away from the emerging surface of the lens, this kind of super-resolution imaging scheme is meaningful for practical applications.
If we observe the spatial spectrum on a plane, we will find that the observed results will deviate from the theoretical results in a great degree, which is easy to be comprehensible considering the focusing behavior of the lens for different oblique incidences. Hence, even though high spatial frequency information can also be detected on the positions always from the focuses, reconstructing the subwavelength images on a plane is still troublesome, which may need complex post-processing.
4.2 Double-lenses super-resolution imaging
Even though the long-distance super-resolution imaging can be realized just by a single hyperbolic GRIN lens, the imaging speed is not favorable because of the time-consuming point by point detection and the post-processing (including the IFT and the adjustment of the amplitude and phase). Our further studies show that the detection and post-processing can be avoided just by using another lens.
We simply describe the idea as follows: first, the enhanced spatial FT is operated on the subwavelength excitation sources by the first lens; next, the second lens is put in the focal “plane” to receive the spectrum information and do spatial IFT. Because the focuses of the first lens are not in a plane, the input surface of the second lens is designed as “V” shape to coincide with the observation path given above; and last, the super-resolution images emerges near the output surface of the second lens. Here, we use the two lenses with the same electromagnetic parameters to do the spatial FT and IFT, respectively. This is reasonable because the restored signal f2(y) from twice consecutive spatial FT is the even symmetric function of the original signal f1(y), i.e. f2(y) = f1(–y). The schematic is shown in Fig. 6(a). The width of the second lens is equal to the first lens. The maximum and minimum thickness of the second lens is 7λ and 3λ, respectively. Here, arc chamfering with the radius being 1.5λ is operated on each of the corners.
The same as in Fig. 5(b), the excitation source consists in two rectangular pulses, which are λ/15 away from the input interface of the first lens. The interval and width of the two pulses are λ/5 and λ/15, respectively. We compare the theoretical result and the simulated result along the emerging surface of the second lens. The two results are well matched, as shown in Fig. 6(b). The amplitude distribution of the magnetic field is also given inset of the figure. From the enlarged view of the inset figure, the images of the two subwavelength pulse are clearly resolved. So the resolution of λ/5 is also demonstrated. Even though the images after the second lens can be only resolved in the near filed, the second lens which can be regarded as a detection device such as a camera, is actually optically long-distance away from the first lens (the maximum and minimum distances are 7λ and 3λ, respectively).
Note that, the spatial FT and IFT, the amplitude and phase adjustment is all done based on hardware, so real-time imaging is realized. Except for the competitive imaging speed, the double-lens super-resolution imaging scheme has the following advantages: (a) the flat input surface is convenient for targets detection; (b) because of the non-overlap spatial spectrum, the super resolved image is reconstructed with just one time illumination. And there is no need of any algorithms during the reconstruction .
5. Artificial construction of the lens
5.1 How to obtain a super NA
In this part, we will show that the extraordinary NA of the lens can be practically realized by just utilizing low index materials. Here, the gradient index of the lens is obtained by using multi-layer hyperbolic metamaterials . Each of the unit layers of the metamaterial is composed by two-layer different materials with positive and negative permittivity, respectively. When the thicknesses of the layers are much smaller than the wavelength, the effective relative permittivity of the metamaterial can be expressed by 
When using silver to construct the lens, we have ε1 = –3.99 + 0.01i. Substituting εx = −1, εy = 2 + 0.069(y/λ)2 and the value of ε1 into to Eq. (8), we can get the values of f0 and ε2, which are plotted in Fig. 7(a). Note that, the maximum value of ε2 is 2.38, which is much smaller than the square of the NA (i.e. NA2 = 3.06ε2 for this case). It indicates that a super NA can be obtained by constructing the lens in this way.
The spatial-varying ε2 can be realized by perforating subwavelength holes arrays in glass. The relative permittivity of glass is εg = 2.41. This kind of fishnet metamaterial has been widely used in transformation optics . The effective permittivity of the fishnet metamaterial can be expressed by ε2 = f + εg (1−f), in which f is the volume fraction of the holes . So we can adjust the value of ε2 by changing the radius of the holes. The schematic of the unit layer is shown inset of Fig. 7(a).
5.2 Verification of simulation results
So we can construct the lens by combining the hyperbolic metamaterial and the fishnet metamaterial. When numerically verifying the effectiveness of the constructing method, we uniformly divide the whole lens into 300 unit layers. The thickness of the lens is 14λ. Hence, the thickness of each unit layer is Δd = 14λ/300. Then the thickness of the silver in each of the unit layers is determined by Δds = Δd·f0. The other parameters keep same as above. The results of the numerical experiments are shown in Figs. 7(b)-7(d), corresponding to the incidences with ky /k0 = 0, 0.5, 1.5. From these figures, the internal and external focusing can be also observed. Note that, both of the focal positions are slightly different with the results of the theoretical model in Fig. 3. This is easy to be explained. In Fig. 3, the electromagnetic waves are continuously manipulated in the metamaterial, however, in Fig. 7, they are only manipulated at the interfaces of each layers. In fact, the energy flows in the combined metamaterial via the coupling of surface plasmon polaritons (SPPs) . Decreasing the thickness of the unit layers will shorten the longitudinal (x-direction) coupling length from one layer to the adjacent layer. So the differences can be reduced by increasing the number of unit layer.
We also demonstrate the super-resolution imaging of the constructed lens by simulation. Here, the amplitude and phase adjustment and the spatial IFT are operated by the theoretical lens introduced in the part of 4.2. This is effective and convenient for us to assess the ultimate performance of the constructed lens. As illustrated in Fig. 8, the two subwavelength images separated by λ/5 are also clearly resolved. The amplitude distribution of the magnetic field and the enlarged view of the images are given inset of Fig. 8.
6. Discussion and conclusions
Note that, the idea presented in this paper is not restricted by the working frequency, which also means that our idea can be extended to the terahertz or microwave frequency band. Considering the advantages of our proposed super-resolution imaging scheme and the double-lenses system configuration, our work may open up a new avenue for designing high-speed microscopes. Distinguished to the conventional techniques , this kind of microscopic scheme can meet the demand of the resolution improvement and the working distance increasing at the same time.
Even though the input spatial frequency bandwidth of the spatial FT is just extended to 2.7k0 here, it can be further enhanced theoretically by adding the width of the lens (the discipline of the permittivity distribution remains unchanged). So the imaging resolution of the lens can be also further enhanced theoretically.
In conclusion, we have designed a plane hyperbolic GRIN lens with a super NA. Due to the momentums compression in the GRIN hyperbolic metamaterial and the negative refraction at the interface of the metamaterial and air, evanescent wave with the transverse momentums up to 2.7k0 can be focused over optically long distances, which is verified by both the ray tracing and the full-wave simulation. Assisted by the enhanced spatial FT of the lens, we proposed a new long-distance and high-speed super-resolution imaging scheme and the super resolved capability of λ/5 is demonstrated.
This work is sponsored by the National Natural Science Foundation of China under contract 61471007, and the Fundamental Research Funds for the Central Universities (“985 Project”).
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