## Abstract

The subwavelength imaging phenomenon in Maxwell’s fisheye lens with one drain has been reported previously. In this paper, we theoretically find that coherent perfect absorbers (CPAs) perform well in generalized Maxwell’s fisheye (GMFE) lenses. Such CPAs are embedded inside the GMFE lenses to absorb the incoming coherent waves. They can be served as drains and dramatically improve the resolution of images in the GMFE lenses. In particular, they can be applied to realize the subwavelength imaging. We also study the multiple imaging characteristics of GMFE lenses with several CPAs in wave optics. Full-wave simulations were performed to verify the imaging functionalities.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Transformation optics [1–3] is an efficient tool to manipulate the propagation of light to design various optical devices, such as cloaks [4–6], field concentrators [7,8], rotators [9,10] and imaging devices [11,12]. Maxwell’s fisheye lens [13] is an absolute instrument [14–18] that any source inside the lens will induce images or self-images with no aberration, which has important applications in integrated photonics [19,20]. GMFE lens, whose refractive index profile can be obtained by a general method for designing absolute instruments [16]. Both of them are related to another important lens, the Mikaelian lens whose refractive index profile is 1/cosh(x). It has been recalled for the imaging functionalities along lines [17,21]. Researchers have established the connection between Mikaelian lens and Maxwell’s fisheye lens by utilizing transformation optics (conformal mapping) and found several interesting imaging effects along curves [22].

Coherent perfect absorbers [23–25] (CPAs), which is also known as time-reversed lasers, received significant attention in recent years. CPAs can absorb electromagnetic radiation completely by controlling the interference of incident waves [25]. It has been demonstrated that CPAs can be realized by using planar structures [23,26,27], PT-symmetric structures [28], guide-mode structures [29] and other complex structures [30]. Further results on CPAs include perfect absorption of TM-polarized modes by metallic structures, and coupling to the surface plasmons [31], excited inside a 2D cavity with absorbing circular boundary [32] and other fields beyond the optics, including acoustics [33–35], polaritonics [36] and quantum optics [37].

Perfect imaging in Maxwell’s fisheye lens [38,39] has been discussed and demonstrated for wave optics recently. It has been proved that Maxwell’s fisheye lens can produce perfect imaging by a perfect drain added at the image position for a converging wave [38,40,41]. The previous work has shown that subwavelength CPA performs well as a drain in Maxwell’s fisheye lens [42]. These CPAs can be practically designed by utilizing composite metamaterials [43] with narrow bandwidth. But it is still unclear that the drain performance of a CPA or multiple CPAs embedded in GMFE lens. In this article, we will simply start from the design of subwavelength CPAs. Later on, we embed the desired CPAs into GMFE lenses with different parameters and explore the imaging functionalities numerically. The comparison with GMFE lens without CPAs will also be displayed. All the imaging functionalities will be confirmed by full-wave simulation from commercial software COMSOL.

## 2. CPAs in generalized Maxwell's fisheye lenses

To begin with, let us study how to design a desired CPA for GMFE lens. As shown in Ref. [42], we have known the general method of designing subwavelength CPA for Maxwell’s fisheye lens by solving the scattering equations based on Bessel and Hankel functions. Here, we can simply obtain the parameters of CPAs by applying this method. For GMFE lens, the refractive index profile is given as [44]

where $r = \sqrt {{x^2} + {y^2}}$ and*m*is a parameter that can be any real number. The imaging characteristics of GMFE lens vary with the value of

*m*in geometrical optics. For instance, the GMFE lens will have no self-imaging effect if

*m*is irrational number [22]. In this article, we take

*m*as 1, 1.5 and 2 respectively and the images of each GMFE lens is 1, 2, 3, which is typical in our system. We set that the radius of GMFE lens is unity and the boundary is a perfect electric conductor (PEC) (mirror in geometrical optics). The GMFE lens is only valid for transverse electric polarized waves, so we set

*μ*= 1 for the system. The parameters of CPAs contain radius

*R*and complex refractive index

*n*where we set

*R*= 0.01m (subwavelength size compared with the wavelength, which is 3m in our system, the unit here could be arbitrary), and the real and imaginary part of

*n*can be obtained by solving the equations including electric field distribution, boundary conditions and the definition of CPA and drawing the contour map of the solutions. Then, we embed the CPA into a background material with a permittivity ${\varepsilon _{0}} = {n}_{0}^{2}$. In general, we plot the ray trajectories of GMFE lens with

*m*= 1, as shown in Fig. 1(a). If we put a source at ($- \sqrt {2} $/4, $\sqrt {2} $/4), it will have a perfect image at ($\sqrt {2} $/4, $- \sqrt {2} $/4) where we put CPA there. If

*m*= 1.5, we will have two locations for CPA placements (if

*m*= 2, we have three locations), as shown in Fig. 1(b) (and Fig. 1(c) for

*m*= 2). The source and images in GMFE lens are on the circle of radius 0.5 meter. The CPAs located at the image positions and the blue curve in the right part of the schematic plotted in Fig. 1(d) represents the amplitude of the electric field on the black dotted circle that start from (−0.5,0) and was plotted counter clockwise.

## 3. Discussion and comparison

#### 3.1 Maxwell’s fisheye lens with one CPA

For *m* = 1, it will come back to the form of Maxwell’s fisheye lens. Researchers have shown that perfect drains performed well in Maxwell’s fisheye lens. Here, we verify it using our system. The CPA here has a complex refractive index *n *= 32.827 + 6.5412*i* and the background material has a permittivity ${{\varepsilon }_{0}} = $ 2.56. Firstly, we calculate electric field of GMFE lens without CPA. Full wave simulation is given in Fig. 2(a) and we calculate the amplitude of electric field on the circle with a radius of 0.5m where the source and CPAs located, as shown in Fig. 2(b). The perfect imaging functionality of Maxwell’s fisheye lens in geometrical optics disappears in wave optics. The electric field energy spread till the whole area distributed normally near the image point obtained from geometrical optics without a good convergence in Fig. 2(a). Then, we embed a CPA at ($- \sqrt {2} $/4, $\; \sqrt {2} $/4) where the image is located. It is interesting that the energy generated by the line-current source will converge at the CPA position in Fig. 2(c), leading to a sharp resolution about 0.0197*λ* by calculating FWHM (full width at half maxima) for imaging functionality of Maxwell’s fisheye lens compared to the one without a CPA, as shown in Fig. 2(b) and Fig. 2(d).

It has been shown that misplacing the active drain a bit does not destroy the peak completely, but a significant portion of the energy can be extracted even in the neighborhood of the correct image position [45]. Here, the CPAs in our system can be regarded as passive drains [40]. However, it is also necessary to explore the behavior of CPA shifting from the image position. We simply move the position of CPA on the circle with a radius of 0.5m and plot all the amplitude of electric field in one figure as shown in Fig. 1(e). We put source at 7*π*/4 position and the corresponding image locates at 3*π*/4 position. We move the CPA from *π*/4 to 5*π*/4 and mark all the CPA positions’ peaks with red pentagram, connect them with a red line. Furthermore, we calculate the FWHM as a function of angular coordinate *θ*, and find that the maximum is around 3*π*/4 position which is located at the image position after eliminating errors in numerical calculations.

#### 3.2 GMFE lenses with multiple CPAs

Since CPA is an almost perfect drain for Maxwell’s fisheye lens, which could in principle be applied into GMFE lenses that have more than one image, we will test the CPA with two GMFE lenses and different numbers of CPAs will be embedded into GMFE lenses for further comparison. Firstly, let us set *m* = 1.5 for GMFE lens, which has two images. Here, the CPA’s complex refractive index *n *= 32.007 + 6.0331*i* and the background material has a permittivity ${{\varepsilon }_{0}} = $ 1.58. If we put a source at ($- \sqrt {2} $/4, $\; \sqrt {2} $/4), it will cause images at ($- \textrm{sin(}\frac{\mathrm{\pi }}{{\textrm{12}}}\textrm{)}$/2, $- \textrm{cos(}\frac{\mathrm{\pi }}{{\textrm{12}}}\textrm{)}$/2), ($\textrm{cos(}\frac{\mathrm{\pi }}{{\textrm{12}}}\textrm{)}$/2, $\textrm{sin(}\frac{\mathrm{\pi }}{{\textrm{12}}}\textrm{)}$/2). If we embed one CPA, either image is reasonable due to the symmetry of GMFE lens. In general, we first calculate the GMFE lens without CPA, as shown in Fig. 3(a) and Fig. 3(b), representing the electric field distribution of GMFE lens and the amplitude of electric field on the source/image circle respectively. The electric field excited from the source distributes normally in space, even without any increasement near image points. Then we embed a CPA at ($- \textrm{sin(}\frac{\mathrm{\pi }}{{\textrm{12}}}\textrm{)}$/2, $- \textrm{cos(}\frac{\mathrm{\pi }}{{\textrm{12}}}\textrm{)}$/2) and curious phenomenon occurs: the electric field converge at the CPA position that behaves like a drain, as shown in Fig. 3(c), while the one without CPA doesn’t have field converging. The amplitude of the electric field on the circle plotted in Fig. 3(d) also verify our assumption that CPA still works to enhance the resolution of GMFE lens dramatically without any degeneration. Since single CPA has shown great functionality in GMFE lens, it is feasible to presume that two CPAs will still perform as perfect drains. Then, we embed two CPAs at the image points of GMEF lens and full wave simulation is shown in Fig. 3(e). The enhancements of electric field occur as well at both image points with the help of CPAs, which is slightly weakened compared to one CPA due to the conservation of energy in space. Besides, we can also find that the two peaks are symmetric about the middle peak in Fig. 3(f), which is consistent with the imaging characteristics in geometrical optics. The results above have shown that CPAs are almost perfect drains for this GMFE lens and it is rational to extend them to other GMFE lenses.

Finally, we set *m* = 2 for GMFE lens and the number of images for this lens is three. The CPA here has a complex refractive index *n *= 31.146 + 5.5106*i* and the background material has a permittivity ${{\varepsilon }_{0}} = $ 0.89. Thus, we have more choices for the placement of CPAs and more curious phenomenon can be explored. In general, we put a source at ($- \sqrt {2} $/4, $\; \sqrt {2} $/4) and three images will locate at ($\sqrt {2} $/4, $- \sqrt {2} $/4), ($- \sqrt {2} $/4, $- \sqrt {2} $/4), ($\sqrt {2} $/4, $\; \sqrt {2} $/4) respectively. For one CPA, we have two options. One is to put at ($\sqrt {2} $/4, $- \sqrt {2} $/4) which locates at the opposite side of the source, the other is to put at either image close to the source due to the symmetry of GMFE lens. Firstly, full wave simulation of GMFE lens without CPA is shown in Fig. 4(a) and the amplitude distribution on the circle is shown in Fig. 4(b). Electric field excited from the source and spread till whole space without any convergence at image points. Then, we embed a CPA at ($\sqrt {2} $/4, $- \sqrt {2} $/4) and electric field simulation is shown in Fig. 4(c). The enhancement occurs as well which is similar to the MFE lens with one CPA in this case, while the amplitude of electric field at the CPA position degenerate slightly, which decrease the resolution (about 0.0216*λ* here), as shown in Fig. 4(d).

Next, we embed the CPA at ($- \sqrt {2} $/4, $- \sqrt {2} $/4) that close to the source. It is no surprising that CPA enhances the resolution of image, as shown in Fig. 4(e), which is comparable to the GMFE lens with *m* = 1.5. We can verify it by contrasting with the amplitude of electric field on the source/image circle shown in Fig. 3(d) and Fig. 4(f), the imaging resolution shown in Fig. 4(f) (about 0.0207*λ*) is slightly smaller than that in Fig. 3(d) (about 0.0201*λ*), which can be attributed to the type of GMFE lens. Hence, it can be considered that CPA is an almost perfect drain for GMFE lens. To prove our assumption, further research that embed more CPAs into GMFE is needed. Firstly, we embed two CPAs into GMFE with *m* = 2 and two cases are included, one is embedded at ($- \sqrt {2} $/4, $- \sqrt {2} $/4), ($\sqrt {2} $/4, $\; \sqrt {2} $/4) which is symmetric about the source and the other is embedded at ($\sqrt {2} $/4, $- \sqrt {2} $/4), ($- \sqrt {2} $/4, $- \sqrt {2} $/4) which is asymmetric. We also plot both the electric field as shown in Fig. 4(g) and Fig. 4(i), and the amplitude distribution along the source/image circle, as shown in Fig. 4(h) and Fig. 4(j). It can be easily seen from these simulations that CPAs function as well. The former one with symmetric CPAs also perform symmetric electric field and the amplitude of the electric field at CPA positions is identical which can be further proved in Fig. 4(h). For the latter one with asymmetric CPAs, the enhancement of electric field at CPA positions degenerate slightly as the distance increases. Last but not least, we consider to embed CPAs at all image positions. It is quite similar to the combination of two cases described above that the electric field of CPAs located near the source are symmetric and the resolution of these positions is higher than the deviant one, as shown in Fig. 4(k) and Fig. 4(l).

## 4. Conclusions

In conclusion, we theoretically prove that the designed CPAs embedded in GMFE lenses with different parameters perform drain roles very well. The enhancement of electric field at the CPA position occurs in all cases but with a different degree, i.e., the GMFE lens with one CPA has higher resolution than the one with one more CPA and different CPAs in one GMFE lens may have divergent resolutions, which can be attributed to the distance between CPA and source, and the conservation of energy in space. Such a system will be very useful in multiple point subwavelength imaging designs. In addition, such CPAs can be used in other lenses or imaging devices that perform other special imaging functionalities. For examples, we can design a system based on more complex GMFE lenses or other absolute instruments, including the Luneburg lens [46], the Miñano lens and the Lissajous lens. It would be very interesting to demonstrate the same imaging properties in each system. Such gradient refractive index lenses will have important applications in silicon waveguide devices [47,48] in the future.

## Funding

National Natural Science Foundation of China (92050102, 11874311, 61675173); Fundamental Research Funds for the Central Universities (20720200074).

## Disclosures

The authors declare no conflicts of interest.

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