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Abstract
In order to realize super-resolution imaging of point source at any positions within a large object distance range, a graded-index equivalent medium (GEM) flat lens, which can break through the object distance limit d (d is the lens thickness), is analyzed by negative refraction. Based on this analysis, graded-index photonic crystal (GPC) flat lens with a large object distance is designed. Its imaging resolution can reach up to 0.4λ at the maximum object distance of 5d, which breaks through the diffraction limit.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The negative refractive material (NRM) is a kind of material that has the negative permeability and negative permittivity simultaneously [1–4]. In the NRM, the propagation direction of the electromagnetic wave is opposite to that of the energy, i.e., the evanescent wave can be enhanced when it transmits inside the NRM [5,6]. Thus the imaging resolution of the flat lens prepared by the NRM can break through the diffraction limit (such lens can be named as superlens) [7]. Generally, the NRM is constructed with metal structure, however, it is unsuitable for practical applications due to its high absorption loss in the optical frequency [4]. On the contrary, because of the qualified negative refraction effect and low-loss, dielectric-based photonic crystal (PC) is regarded as the ideal materials for realizing the superlenses [8,9].
PC, an artificial micro-structure whose dielectric constant is arranged in period [10,11] or quasi-period [12] and the lattice constant is on the order of light wavelength. Owing to the characteristics of photonic band gap [10,13], photonic localization [11,14] and negative refraction [8,9], PC can provide wide prospect in optical communication and optical integrated systems including fibers [15–17], waveguides [18,19], filters [20,21], sensors [22], prisms [23], and flat lenses [14,24–38]. In the field of PC, the focusing and imaging of flat lens have become a hot research topic nowadays, but we still face the problems that the common photonic crystal (CPC) (including periodic and quasi-periodic CPCs) flat lens cannot be used to focus the plane wave to a spot and have very limited object distance for point source imaging. The point source imaging in the periodic CPC flat lens needs the light source close to the lens, i.e., u = a [24], where u is the object distance, a is the lattice constant; or follows u + v = d [26,27], where v is the image distance, d is lens thickness. The object distance of point source imaging for quasi-periodic CPC flat lens is half of the lens thickness, i.e., u = d/2 [33–36,38]. Its object-image relationship satisfies u + v ≈d [33], or u + v < d [34]. Although plane wave focusing has been achieved by using GPC flat lens [28–30], the object distance for point source imaging is still consistent with CPC flat lens [31,32]. Obviously, the short object distance of the CPC severely limits its applications especially in biological structure imaging which must be based on non-contact and scatheless measuring. Hence, enlarging object distance of the PC flat lens with super-resolution is a very urgent and meaningful work. According to negative refraction [26,27], PC flat lenses with different effective refractive indices (ERIs) correspond to different imaging object distance ranges, and the object distance limit of PC flat lens can be broken through by appropriately designing the ERI. Moreover, the ERI of the PC flat lens can be controlled by its material or structural parameters, so the specially designed GPC flat lens can break through the object distance limit theoretically.
In this paper, a GEM flat lens which can break through the object distance limit is analyzed by negative refraction. Based on this analysis, GPC flat lens with large object distance is designed. Its imaging resolution can reach up to 0.4λ at the maximum object distance of 5d, which breaks through the diffraction limit.
2. Model and theory
According to the theory of equal-frequency line (EFL) [39], a CPC that near its band gap can be equivalent to a homogeneous medium which possesses a certain ERI when its EFL (named equal-frequency surface in three-dimensional case) is isotropic. It can be used to image the point source when the ERI is negative, as shown in Fig. 1.
It can be seen from Fig. 1(a) that the CPC flat lens has an object distance limit, and the point source cannot be imaged when the object distance exceeds the critical value. According to negative refraction, the PC flat lens with neff = −1 can image within the object distance range (0, d) (d is the lens thickness), that is, the position of the point source P∈ (-d, 0) [26,27]. The larger the object distance, the smaller the value of the incident angle i to the flat lens, and the smaller the value of the refraction angle t, so the lower the probability that the beam converges inside the flat lens, that is, the lower the possibility of imaging. So if to realize point source imaging with large object distance, it is necessary to increase the refraction angle. When the ERI of the lens is large than −1 (such as −0.7), the refraction angle can be increased to realize the point source imaging with large object distance, as shown in Fig. 1(b). Nevertheless, in this case the imaging efficiency of the point source with small object distance is low or even cannot be imaged due to the total reflection.
Although different ERIs can be achieved by changing the frequency of incident light source or the structure of the PC, the range of imaging object distance is small, which cannot meet the actual demand. Therefore, it is necessary to design a structure that can break through the object distance limit of the PC flat lens. It can be seen from Fig. 1 that the point source imaging of CPC flat lens with different ERIs has different object distance ranges, so negative refractive index with gradient arrangement for the PC can break through this object distance limit. To this end, a GEM flat lens is designed, its structure and partial refractive indices distribution are shown in Fig. 2.
In Fig. 2(a), the lens thickness d = 10 μm, the refractive index in the middle part is −1, and it increase to −0.1 with the gradient Δn = 0.1 along both sides of the X-axis gradually. S, I, u and v represent the point source, image point, object distance and image distance, respectively. Since the lens is no longer a homogeneous medium, the incident light will be refracted multiple times inside the lens until it exits from the other side of the lens. In view of the beam that incident at a small angle contributes a lot to the imaging, the middle parts of the GEM flat lens are analyzed. In Fig. 2(b), the absolute value of the negative refractive index gradually decreases from the middle line to both sides, so the beam incident with larger angle corresponds to the smaller absolute value of negative refractive index. According to the distribution of negative refractive indices shown in Fig. 2(b), the light beam is refracted continuously inside the lens, so the point source can be imaged with a longer distance, and in turn breaks through the object distance limit.
In order to verify the above analysis, the imaging effects of a homogeneous equivalent medium (HEM) flat lens with refractive index n = −1 and a GEM flat lens are simulated with u = 0.5d, 2d, 5d, respectively. In order to compare with the subsequent simulation results of PCs, the incident frequency is set to 0.313, the beam polarization is assumed to be TM mode, and the background material is assumed to be air. To prevent interface reflection, all boundaries are set to perfect matching layers. The simulation results are shown in Figs. 3(a)-3(c). When HEM flat lens’ refractive index n = −1, it can image a point source for u = 0.5d, but has no imaging effect for u = 2d, 5d. The imaging range can be significantly improved by the GEM flat lens as shown in Figs. 3(d)-3(f). For all the object distances u = 0.5d, 2d, and 5d, the GEM flat lens has good imaging effects, which are consistent with the theory analysis.
Fig. 3 Imaging effect of the HEM flat lens with refractive index n = −1 for the incident point source at different positions: (a) u = 0.5d; (b) u = 2d; (c) u = 5d. Imaging effect of the GEM flat lens for the incident point source at different positions: (d) u = 0.5d; (e) u = 2d; (f) u = 5d.
Based on the above analysis of GEM, a two-dimensional triangular lattice GPC flat lens based on EFL theory is designed as shown in Fig. 4. The dielectric cylinders are arranged with a triangular lattice in air (nair = 1). The parameters of the flat lens are as follows, the thickness d = 10 μm, the width w = 34 μm, the radius of scatterer r = 0.44a (a = 1 μm), the refractive index of middlemost row scatterers n0 = 3, and the refractive indices of other scatterers gradually increase along both sides of the X-axis (the darker the color of the scatterers, the higher the refractive index). The beam propagates along the X-axis direction. Si represents the point source at different positions, Ii represents the corresponding image point, and ui and vi represent the object distance and image distance respectively, i = 1, 2, 3.
In PCs, the basic principle of negative refraction is to create a “negative group refractive index” using the special dispersion relationship of PCs at the bandgap edge [40]. The relationship between the group velocity vector vg and the wave vector k is as follows [39]
Therefore, the direction of light refraction in the PC is determined by the EFL theory [39].Figure 5(a) shows the incident direction of the input beam in air and two different refractions of the beam incident from the air into the PC. According to negative refraction, the directions of the wave vector and the group velocity should be opposite, so vg1 is the negative refractive group velocity. Owing to the GPC scatterers with gradient refractive indices distribution, the plane wave expansion method cannot be used to obtain the EFLs directly. Thus, the band structures of GPC should be calculated from the band structures of CPCs with different refractive indices respectively. The band structures of the GPC are obtained by calculating the TM band structures of CPC whose refractive indices of scatterer correspond to the middlemost row scatterer with n = 3 and the outermost row scatterer with n = 3.495 of GPC, as shown in Fig. 5(b). It can be inferred from Fig. 5(b) that, when the ERI is −1 in the CPC where the refractive index of scatterer is 3, the normalized frequency is 0.313, and the refractive index of the scatterer is 3.495 for the CPC with an ERI of −0.1 at the normalized frequency of 0.313. In order to make it equivalent to GEM flat lens, the GPC transforms the gradient by taking two rows of scatterers as units, and the gradient index Δn = (3.495 - 3.000)/9 = 0.055 where the 9 is the number of refractive index layers of the GEM flat lens. At the same time, since the GPC is a triangular lattice structure with only one row of scatterers in the middle, a row scatterer is added to each side to coincide with the GEM flat lens.
Fig. 5 (a) Analysis of beam propagation direction using EFL; (b) Partial GPC band structures diagram.
3. Results and discussions
According to the results shown in Fig. 3, the GEM flat lens can realize point source imaging with large object distance. Meanwhile, the GPC flat lens can coincide with the GEM flat lens based on the theoretical analysis shown in Figs. 4 and 5. Therefore, the point source imaging with large object distance can be realized by the GPC flat lens. In order to verify the above analysis, the CPC and GPC flat lenses are analyzed under the same conditions as that of the GEM flat lens, and the results are shown in Fig. 6.
Fig. 6 Imaging effect of the CPC flat lens with a refractive index n = 3 for the incident point source at different positions: (a) u = 0.5d; (b) u = 2d; (c) u = 5d. Imaging effect of the GPC flat lens for the incident point source at different positions: (d) u = 0.5d; (e) u = 2d; (f) u = 5d.
It can be seen from Fig. 6 that the CPC flat lens has an object distance limit. At u = 0.5d, the CPC flat lens has good imaging effect; but it cannot be imaged for u = 2d and 5d, which are consistent with Figs. 3(a)-3(c). GPC flat lenses have good imaging effects for all these object distance u = 0.5d, 2d and 5d. These results are equivalent to Figs. 3(d)-3(f). In order to show that the GPC flat lens can image at any positions within the object distance range. The imaging characteristics of the GPC flat lens with different object distances in the range u ∈ [0.5d, 5d] (i.e. u ∈ [5 μm, 50 μm]) are calculated respectively, which are shown in Fig. 7.
Fig. 7 The relationship between the imaging characteristics of the GPC flat lens and the object distance: (a) the intensity ratio between the image intensity and point source intensity; (b) the image distance and the FWHM.
It can be seen from Fig. 7(a) that, the GPC flat lens can image the point source, the ratio of the imaging intensity to the source intensity fluctuates around 15‰. In other words, the decrease in imaging intensity is not sharply with the increase of object distance. These results indicate that the large object distance GPC flat lens has high practicability. From Fig. 7(b), as the object distance increases, both the image distance and the full width at half maximum (FWHM) fluctuates first (u < 1.5d) and then changes with tendency of decrease. It can be inferred from this tendency, when u > 1.5d, the image quality increases with the increase of object distance. However, the corresponding image intensity and the image distance become smaller with the increase of object distance. When u > 2d, the imaging of GPC flat lens breaks through the diffraction limit, and the imaging resolution can reach up to 0.4λ at the maximum object distance of 5d.
In the imaging system, the numerical aperture (NA) of a lens is defined as
where n is the refractive index of the medium between the lens and the point source (n = nair in this paper), θ is aperture angle, as shown in the Fig. 1(a).In the GPC flat lens as shown in Fig. 4, Eq. (2) can be rewritten as
where u1 is the object distance, w is the width of flat lens. In Eq. (3), n = 1 and w is fixed for the GPC flat lens, so the NA decreases as u1 increases, that is, NA < 1. Meanwhile, the criterion of diffraction limit is defined as 0.5λ/NA [41], so the GPC flat lens realize super-resolution imaging when u > 2d.In addition, the NA represents the light-harvesting capability and resolution of the lens, the smaller the numerical aperture, the worse the light-harvesting capability and resolution [42]. In GPC flat lens, the NA decreases with the increase of object distance, but FWHM does not increase with the increase of object distance, which proves that the resolution is not affected by the increase of the object distance and has high stability. The results provide a longer focal depth of object plane compare with the CPC flat lens, which have high application potentials in optical detection [43].
In order to show that the GEM and GPC flat lenses can realize off-axis point source imaging with large object distance, The GEM and GPC flat lenses are analyzed with two off-axis point sources of large object distance, and the results are shown in Fig. 8.
Fig. 8 The off-axis point source imaging with large object distance: (a) GEM flat lens; (b) GPC flat lens.
From Fig. 8, it can be seen that, the GEM and GPC flat lenses can realize off-axis point source imaging with large object distance. The imaging of conventional lens follows the imaging formula of thin lens [44], that is, 1/u + 1/v = 1/f, where u, v and f represent object distance, image distance, and focal length, respectively, so the images of conventional lens are inverted. However, the imaging of negative-refraction flat lens follows a different imaging rule, u + v = d, which implies the images of negative-refraction flat lens are not inverted [45], so Image1 and Image2 are the images of Source1 and Source2 in the GEM and GPC flat lenses respectively, as shown in Fig. 8. Notably, since u + v = d, even the two point sources are located on different transverse planes, the relative distance between of the two images in the negative-refraction flat lens remains constant [45], but the relative distance between of the two images inevitably decreases in the GEM and GPC flat lenses as can be seen from Fig. 8, which is caused by the gradient distribution of ERI. Therefore, these lenses have the function of reducing the relative distance, which is a very interesting phenomenon and will expand the application field of negative-refraction flat lens.
4. Conclusions
Based on negative refraction, the imaging of GEM flat lens is analyzed. The research results indicate that GEM flat lens can break through the object distance limit. Based on this analysis, a GPC flat lens with large object distance is designed. It can realize super-resolution imaging when the object distance ranges from 2d to 5d. The highest resolution reaches up to 0.4λ at the maximum object distance of 5d, which breaks through the diffraction limit. The FWHM does not increase with the increase of object distance, which has high application potentials in optical detection. Besides, the GPC flat lens can realize off-axis point source imaging with large object distance, and can reduce the relative distance of different point sources. The results will provide a very attractive alternative solution for fabrication the optical integrated devices with large object distance and compact size.
Funding
National Natural Science Foundation of China (61405058, 61605166); Natural Science Foundation of Hunan Province (2017JJ2048, 2018JJ3514); Fundamental Research Funds for the Central Universities (531107050979).
Acknowledgments
The authors acknowledge Prof. J. Q. Liu for software sponsorship, and sincerely thank Dr. D. L. Tang for his insights offered in the deep discussions.
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