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Abstract
We demonstrate an isotropic device called 540-degree deflecting lens, which has symmetric refractive index and can deflect parallel beam by 540 degrees. The expression of its gradient refractive index is obtained and generalized. We discover it’s an optical absolute instrument with self-imaging characteristic. Using conformal mapping, we deduce its general version in one-dimensional space. We also introduce a combined lens called the generalized inside-out 540-degree deflecting lens similar to the inside-out Eaton lens. Ray tracing and wave simulations are used to demonstrate their characteristics. Our study expands the family of absolute instruments and provides new ideas to design optical systems.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Optical lenses have a long studying history and they have become indispensable optical devices in scientific research and industrial production. Gradient index lenses, which manipulate light by accumulating desired propagating phases, offer the advantages of large bandwidth, ease of preparation, low cost, and have attracted a great deal of interest in theory and engineering studying. For example, the omnidirectional reflection Eaton lens [1], omnidirectional focusing Luneburg lens [2], and beam aligned Maxwell’s fisheye lens are widely used in optical antenna designs and detection [3–9]. Considering the original versions of the above three kinds of lenses with unit circle structure, their corresponding refractive indices would be ${n_E} = \sqrt {\frac{2}{r} - 1} $, ${n_L} = \sqrt {2 - {r^2}} $ and ${n_M} = \frac{2}{{1 + {r^2}}}$, in which r stands for the distance from the origin and is expressed as $r = \sqrt {{x^2} + {y^2}} $. With the development of metamaterials and metasurfaces, the manufactory of those lenses become realistic. More interestingly, the above functional lenses or the same gradient index lenses are absolute instruments (AIs) [10–12], which means that all the points in the lens area can realize perfect imaging without any aberration, in other words, they can provide super-resolution and be applied in imaging or lithography. Apart from these classic devices, Miñano lens provides images in homogenous region with unit refractive index and is called the inside-out version of Eaton lens, which makes it easier to be realized and inspires us to find lenses with similar properties. In addition, related paradigm and general methods to design both symmetric and asymmetric AIs are developed [13–15], which can tutor us to explore more gradient index lenses with different characteristics.
Combined with transformation optics (TO) theory [16–19], the application scenarios of gradient index lenses have been greatly expanded. Based on the form invariance of Maxwell’s equations under coordinate transformation, transformation optics (TO) theory can arbitrarily regulate electromagnetic field based on demands, providing a lot of novel devices such as invisibility cloaks [20–22], field rotators [23], illusion devices [24], perfect absorbers [25], to name a few. A flat plate lens can be transformed from the parabolic lens to achieve focusing [26]. A low-cross multimode waveguide crossing is stemmed from Maxwell’s fisheye lens via conformal mappings [27]. A thin metamaterial lens for wide-beam radiation by embedding a simple source is realized by non-Euclidean transformation [28]. Moreover, similar concepts and methods are also extended to acoustic [29,30] and elastic wave systems [31]. Meanwhile, due to the connection between virtual space and physical space, TO brings another insight to understand the gradient index lenses. For instance, Maxwell’s fisheye lens is converted to Mikaelian lens [32,33], which corresponds to the Lenz potential and modified Pöschl–Teller potential in quantum mechanics [34,35]. And the Eaton lens of Coulomb potential as well as the Luneburg lens of Hooke potential, are conformally related to the asymmetric Morse lens of the Morse potential [36]. By using TO theory, we can transform a two-dimensional lens into its one-dimensional version in order to understand its imaging characteristic and expand its application in waveguide propagation.
In this work, we study a functional gradient index lens which can deflect the incident light by 540 degrees and its inside-out version. Firstly, we obtain the explicit expression of its refractive index distribution based on Tomáš’ theory [13]. Then, by introducing the character parameter a and applying exponential conformal mapping, we develop its general form and find that all the derivative lenses are absolute instruments. The general 540-degree deflecting lens has multiple images, which possesses an interesting property when the character parameter a is a multiple of three. The property is different from the Maxwell’s fisheye lens, Eaton lens or Luneburg lens. Next, we explore combined cases of the general 540-degree deflecting lens, i.e. its inside-out version, by replacing the refractive index distribution in the region r < 1 with another matched distribution. Ray tracing and wave simulations are used to demonstrate the imaging behaviors throughout the paper.
2. Results and discussion
In 2011, Tomáš Tyc described a general method to design absolute instruments based on the Luneburg’s inverse scattering problem in optics [13]. From his theory, the refractive index distributions of symmetric optical absolute instruments satisfy
Tomáš took Miñano lens (inside-out version of Eaton lens) as an example to explain his method specifically, in which a unit circle filled with air is embedded in the region $r \le 1$. Under this condition, $n({r_0}) = {r_0} = {\; }{L_0} = 1$ and there are two turning points ${r_ \pm }$ that satisfy $f({{r_ \pm }} )= {\; }{r_ \mp }$. The relationship between ${r_ + }$ and ${r_ - }$ is
By substituting ${r_ + }$ with $f({{r_ - }} )$, we can obtain
Inverting Eq. (3), ${r_ - }({{r_ + }} )= f({{r_ + }} )$ can be obtained and $f(r )$ is derived as
When m = 2, after combining Eq. (1) and (4), the refractive index of Miñano lens is:
It is obvious that Eq. (3) may not have inverse solutions with m increasing, so only a few values of m can be applied to design AIs with explicit $n(r )$ which provide images in the homogenous regions. Among them, we derive some explicit expressions of $n(r )$ with different values of m in the region $r > 1$ as follows: when $m = 4$, ${n_1}(r )= \frac{1}{2}\sqrt {{b_1} + {c_1}} + \frac{1}{2}\sqrt { - {b_1} - {c_1} + \frac{4}{{r\sqrt {{b_1} + {c_1}} }}} $, ${a_1} = 9r + \sqrt 3 \sqrt {27{r^2} - 16{r^6}} $, ${b_1} = \frac{{2 \times {2^{2/3}}r}}{{{{({3{a_1}} )}^{1/3}}}}$, ${c_1} = \frac{{{{({2{a_1}} )}^{1/3}}}}{{{3^{2/3}}r}}$; when $m = 2$, ${n_2}(r )= \sqrt {\frac{2}{r} - 1} $; when $m = 1$, ${n_3}(r )= {\left( {Q - \frac{1}{{3Q}}} \right)^2},Q = \sqrt[3]{{ - \frac{1}{r} + \sqrt {\frac{{{1^2}}}{{{r^2}}} + \frac{1}{{27}}} }}$; when $m = \frac{2}{3}$, ${n_4}(r )= \frac{1}{{2\sqrt 2 }}{\left( { - 1 + \sqrt {1 + \frac{8}{r}} } \right)^{\frac{3}{2}}}$. It is noted that such AIs can be obtained for any rational m, however, some of them are explicit and simple (such as Miñano lens), some are hard to write down and may require the help of numerical fitting method. Here we only list a few cases of interest. Similar to Miñano lens, their corresponding homogenous region is $r \le 1$ and also filled with air.
To compare and showcase the difference among lenses mentioned above, we list their refractive index distribution $n(r )$, corresponding $f(r )$ and values of m in Table 1. It is obvious that different m and $f(r )$ designed from the theory will decide the function of absolute instruments, which also proves the correctness of Eqs. (1–4).
Table 1. Comparison of Different Absolute Instruments
Considering the inside-out relation between Eaton lens and Miñano lens, we apply the above refractive indices in the region $r \le 1$ only and make sure outsides is air to find other AIs. Interestingly, it is revealed that these AIs can deflect parallel light rays by 90 degrees, 180 degrees, 360 degrees and 540 degrees, separately, corresponding to the refractive indices of ${n_1},\; {n_2},{n_3},{n_4}.$ We call the above eight lenses as deflecting lenses and their inside-out versions of different angles. Their imaging characteristics are shown in Figs. 1(a)–1(h). Among them, the 90-degree, 180-degree and 360-degree deflecting lens are well-known as 90-degree rotating lens, Eaton lens and invisible sphere lens. They can be used as beam splitters or connectors to effectively control the light flow and have become the most representative single isotropic optical structures. Since the 540-degree deflecting lens also has precise and straightforward refractive index distribution like the other three lenses, we focus on its inherent imaging characteristic in order to provide some fundamental theory for further potential applications. Besides, by generalizing 540-degree deflecting lens with a variable parameter a, more AIs and their inside-out versions are found and studied.
Fig. 1. (a)-(d): Parallel rays deflect 90 degrees, 180 degrees, 360 degrees, 540 degrees after passing through unit-circle lenses with refractive index n1, n2, n3, n4 from the air. (e)-(h): Inside-out versions of deflecting lenses (a)-(d) with air embedded in the region $r \le 1$. Red curves and blue points stand for light rays and imaging points, separately. The black circle stands for r = 1.
We notice that 540-degree deflecting lens and Eaton lens perform the same omnidirectional reflection due to the restriction of 2π range in the polar coordinate. The difference is that the ray will experience one more round in the 540-degree deflecting lens and this extra loop is very applicable in time-delay devices similar to the invisible sphere lens. In order to further explore its imaging characteristic, we introduce TO into our research and use logarithmic conformal transformation $w = ln(z )$ to transform the z space ($z = x + iy,\; r = \sqrt {{x^2} + {y^2}} $) into the w space ($w = u + iv,\; {r_w} = \sqrt {{u^2} + {v^2}} $). The coordinates in z-space and w-space are corresponded as below: $x = {e^u}\cos v$, $y = {e^u}\sin v$, ${r_w} = {e^x}$. Based on conformal mapping ${n_z}|{dz} |= {n_w}|{dw} |$, we obtain the refractive index distribution of the corresponding one-dimensional conformal lens of 540-degree deflecting lens as $n(u )= \frac{1}{{2\sqrt 2 }}{\left( { - {e^{\frac{{2u}}{3}}} + \sqrt {{e^{\frac{{4u}}{3}}} + 8{e^{\frac{u}{3}}}} } \right)^{3/2}}$. It is obvious that the light rays emitted from the origin will periodically image along the v direction as shown in Fig. 2(a). The blue dots stand for the self-imaging points and there is an imaging period measured as 6π between every two adjacent dots. Because of the asymmetric refractive index distribution only with respect to u axis, the geometrical ray trajectories are also asymmetric, which differ from those of the Mikaelian lens. To be mentioned, there is a fake imaging point between every two adjacent dots, which is rather deceptive. We then introduce the character parameter a and generalize the variable u to the general form au in the above n(u), and plot the ray trajectories when a = 2 and a = 3 in Fig. 2(b) and (c), respectively. It is found that the self-imaging period is changed into 6π/a as the parameter a scales the ray paths. In the next step, after carrying the inverse conformal transformation $z = \textrm{exp}(w )$, we deduce the general version of the 540-degree deflecting lens in two-dimensional space and call it as the generalized 540-degree deflecting lens, whose refractive index distribution is in the form of
Since the one-dimensional lenses in Fig. 2 are AIs, the rotationally symmetric generalized 540-degree deflecting lenses after conformal transformation are also AIs. According to the conformal transformation, the point (0,0) in the one-dimensional space becomes (1,0) in the two-dimensional space. In Fig. 3, we plot the ray trajectories of the generalized lenses with different parameter a. Figure 3(a) is the 540-degree deflecting lens in Eq. (6) as a = 1, where the rays start from point (1, 0) and pass by the origin 3 times before closing with a single self-imaging point. This is consistent with the imaging period of 6π as shown in Fig. 2(a). When a = 2, as shown in Fig. 3(b), there are 2 images and the trajectories also travel around the origin 3 times. From Fig. 2(b), it can be seen that the imaging period is 3π and it takes 3 cycles around the origin to get accumulated to 6π and obtain 2 images. However, the situation in Fig. 3(c) is a little different. In this case, a = 9/2, the rays pass by the origin twice and then close, but with 3 imaging points, which indicates that the generalized 540-degree deflecting lens has some special and interesting characteristics.
Fig. 2. Conformal 1D lenses of 540-degree deflecting lens with various parameter a in w-space (red curves stand for trajectories while blue points stand for imaging points). Rays start at point (0,0) and present self-imaging along the v direction. (a) a = 1 with imaging period 6π. (b) a = 2 with imaging period 3π. (c) a = 3 with imaging period 2π. In each figure, rays start from the initial point (0,0) in three different directions and the background is the corresponding refractive index distribution n.
Fig. 3. Ray trajectories of the generalized 540-degree deflecting lens accord to different a. The red curves stand for trajectories while blue points stand for imaging points. (a) a = 1 with one self-imaging point. Rays are closed after passing by the origin 3 times; (b) a = 2 with 2 imaging points. Rays are closed after passing by the origin 3 times. (c) a = 9/2 with 3 imaging points. Rays are closed after passing by the origin 2 times; (d) a = 12/5 with 4 images. Rays are closed after passing by the origin 5 times. In each figure, rays start from the initial point (1, 0) in three different directions and the background is the corresponding refractive index distribution ln(n).
After analyses, we conclude that the generalized 540-degree deflecting lens has the following imaging rules when a changes:
To confirm the rules, in Fig. 3(d), we take a = 12/5 as an example, which means p = 12 and q = 5. As the number 12 is not coprime to 15 (i.e., 3q), there will be 4 (i.e., p/3) images and 5 (i.e., q) times travelling around the origin. Taking another example, when a = 3, which means p = 3 and q = 1, the rays will have one self-imaging point and pass by the center one time, which accords with the one-dimensional imaging period of 2π as shown in Fig. 2(c). For negative a, the imaging rules are the same as its counterpart of abs(a).
It is known that Eaton lens as well as the invisible sphere lens have their inside-out versions with a unit circle of air embedded in the region $r \le 1$ and such composite lenses can be used to image in the homogenous region [12,37]. We find 540-degree deflecting lens also has similar properties, which is shown in Fig. 4(a). The rays start from the inner homogeneous air area at point (0.8, 0), then image at the symmetric point (−0.8, 0) and go back to the initial position. Although we expect to find some homogenous medium that can match up with the generalized 540-degree deflecting lens as parameter a changes, it seems like this kind of composite lens will not have closed trajectories or similar imaging properties. Therefore, we replace the inside homogenous area with the matched calculated lens and collectively call this series of lenses as the generalized inside-out 540-degree deflecting lens. This kind of composite lens can overcome the shortcoming of the generalized 540-degree deflecting lens as its refractive index changes rapidly in the region $r \le 1$ and its refractive index distribution is as follows:
When a = 1, it goes back to the embedded air case. We enumerate three cases in Fig. 4(b), (c) and (d) with a = 5/2, a = 3, a = 9/2 respectively.
Fig. 4. Ray trajectories of the inside-out generalized 540-degree deflecting lens accord to different a. The red curves stand for trajectories while blue points stand for imaging points, black circles indicate the matched inner lenses of range $0 < r < 1$ when a changes. (a) a = 1 with 2 imaging points. (b) a = 5/2 with 5 imaging points. (c) a = 3 with 2 imaging points. (d) a = 9/2 with 3 imaging points. All light rays start from the points (0.8,0), and the images are on the circle with a radius of r = 0.8. Background colormaps show the corresponding refractive index distribution n(r).
The imaging property of this inside-out version lens is summarized as:
Fig. 5. The wave simulation pattern of ${E_z}$ amplitude of the inside-out 540-degree deflecting lenses by COMSOL Multiphysics accord to different a. (a) a = 3, (b) a = 9/2. A TE point source is placed at B (0.8, 0). White dashed circles show the inner lenses of r = 1. Simulated frequency is 5 GHz. Related ray trajectories are plotted in Figs. 4(c) and (d).
For the practical fabrication of the above lenses related to 540-degree deflecting lens, it is important to deal with the difficulties caused by the singularities in their inhomogeneous region. Fortunately, there are some available methods to solve this problem. For instance, the inhomogeneous isotropic refractive index can be realized under the guidance of the effective medium theory, such as drilling holes in the dielectric plate [41] or designing metamaterials with different unit cells [4]. The problem of the singularity in the refractive index can be solved by truncating the lens, or by applying some suitable coordinate transformation to replace the singular part with the anisotropic but non-singular medium [42]. Besides, exploring the geodesic lens (the equivalent surface with equal optical path) of 540-degree deflecting lens is a good choice. The geodesic lens will preserve its functionality but only require the homogeneous medium [43–45]. Other methods to realize its fabrication are also expected with the development of material science and process technology.
3. Conclusion
In conclusion, we have studied the 540-degree deflecting lens and its general version and summarize their imaging characteristics by ray tracing and wave simulations in two-dimensional space. Combined with conformal transformation, we get the one-dimensional version of this series of lenses and explain the imaging phenomenon related to three in two-dimensional space. We have also explored the generalized inside-out version of 540-degree deflecting lens and conclude their image patterns. In the future, there are other interesting aspects worth exploring such as the matched geodesic lenses [43,44] and other multifunctional combined lenses based on the generalized 540-degree deflecting lens, as well as their practical applications such as optical connectors, optical phase-delayers and so on. What’s more, the one-dimensional version of the generalized 540-degree deflecting lens can be applied to waveguide transformation and communication system design. In addition, since the Eaton lens, its inside-out version and some superior versions have been manufactured by using metamaterials, metasurfaces or 3D printing technology, we believe our research results can be finally put into application in the future [46–48].
Funding
National Key Research and Development Program of China (No. 2020YFA0710100); National Natural Science Foundation of China (No. 92050102); Fundamental Research Funds for the Central Universities (No. 20720200074, No. 20720220033, No. 20720220134); Shenzhen Science and Technology Program (No. JCYJ20210324121610028).
Acknowledgments
We thank Dr. Yangyang Zhou for the helpful discussions.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. J. Eaton, “On spherically symmetric lenses,” Trans. IRE Prof. Group Antennas Propagat. 4, 66–71 (1952). [CrossRef]
2. R. K. Luneburg, Mathematical Theory of Optics (University of California, 1966).
3. J. Maxwell, “Solution of a problem Camb,” Dublin Math. J 8, 188–193 (1854).
4. Z. L. Mei, J. Bai, T. M. Niu, and T. J. Cui, “A Half Maxwell Fish-Eye Lens Antenna Based on Gradient-Index Metamaterials,” IEEE Trans. Antennas Propag. 60(1), 398–401 (2012). [CrossRef]
5. A. Demetriadou and Y. Hao, “Slim Luneburg lens for antenna applications,” Opt. Express 19(21), 19925–19934 (2011). [CrossRef]
6. M. Liang, W. R. Ng, K. Chang, K. Gbele, M. E. Gehm, and H. Xin, “A 3-D Luneburg Lens Antenna Fabricated by Polymer Jetting Rapid Prototyping,” IEEE Trans. Antennas Propag. 62(4), 1799–1807 (2014). [CrossRef]
7. O. Quevedo-Teruel, J. Miao, M. Mattsson, A. Algaba-Brazalez, M. Johansson, and L. Manholm, “Glide-symmetric fully metallic luneburg lens for 5 G communications at K a-band,” Antennas Wirel. Propag. Lett. 17(9), 1588–1592 (2018). [CrossRef]
8. C. Mateo-Segura, A. Dyke, H. Dyke, S. Haq, and Y. Hao, “Flat Luneburg Lens via Transformation Optics for Directive Antenna Applications,” IEEE Trans. Antennas Propag. 62(4), 1945–1953 (2014). [CrossRef]
9. W. Dong, Y. Lai, and J. Hu, “Detecting spatial chirp signals by Luneburg lens based transformation medium,” Opt. Express 30(6), 9773–9789 (2022). [CrossRef]
10. J. L. Synge, “The absolute optical instrument,” Trans. Amer. Math. Soc. 44(1), 32–46 (1938). [CrossRef]
11. M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Elsevier, 2013).
12. J. C. Minano, “Perfect imaging in a homogeneous three-dimensional region,” Opt. Express 14(21), 9627–9635 (2006). [CrossRef]
13. T. Tyc, L. Herzanova, M. Sarbort, and K. Bering, “Absolute instruments and perfect imaging in geometrical optics,” New J. Phys. 13(11), 115004 (2011). [CrossRef]
14. T. Tyc, H. L. Dao, and A. J. Danner, “Absolute optical instruments without spherical symmetry,” Phys. Rev. A 92(5), 053827 (2015). [CrossRef]
15. T. Tyc and A. J. Danner, “Absolute optical instruments, classical superintegrability, and separability of the Hamilton-Jacobi equation,” Phys. Rev. A 96(5), 053838 (2017). [CrossRef]
16. U. Leonhardt, “Optical conformal mapping,” Science 312(5781), 1777–1780 (2006). [CrossRef]
17. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006). [CrossRef]
18. H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Mater. 9(5), 387–396 (2010). [CrossRef]
19. Y. Liu and X. Zhang, “Recent advances in transformation optics,” Nanoscale 4(17), 5277–5292 (2012). [CrossRef]
20. T. Ergin, N. Stenger, P. Brenner, J. B. Pendry, and M. Wegener, “Three-Dimensional Invisibility Cloak at Optical Wavelengths,” Science 328(5976), 337–339 (2010). [CrossRef]
21. J. Li and J. B. Pendry, “Hiding under the Carpet: A New Strategy for Cloaking,” Phys. Rev. Lett. 101(20), 203901 (2008). [CrossRef]
22. M. Gharghi, C. Gladden, T. Zentgraf, Y. Liu, X. Yin, J. Valentine, and X. Zhang, “A carpet cloak for visible light,” Nano Lett. 11(7), 2825–2828 (2011). [CrossRef]
23. H. Chen, B. Hou, S. Chen, X. Ao, W. Wen, and C. T. Chan, “Design and Experimental Realization of a Broadband Transformation Media Field Rotator at Microwave Frequencies,” Phys. Rev. Lett. 102(18), 183903 (2009). [CrossRef]
24. Y. Lai, J. Ng, H. Chen, D. Han, J. Xiao, Z.-Q. Zhang, and C. T. Chan, “Illusion Optics: The Optical Transformation of an Object into Another Object,” Phys. Rev. Lett. 102(25), 253902 (2009). [CrossRef]
25. Q. Li, Y. Luo, D. Liu, Y. Gao, J. Zhang, L. Ran, and D. Ye, “A Miniaturized Anechoic Chamber: Omnidirectional Impedance Matching Based on Truncated Spatial Kramers-Kronig Medium,” Adv. Opt. Mater. 10(12), 2200381 (2022). [CrossRef]
26. F. Kong, B.-I. Wu, J. A. Kong, J. Huangfu, S. Xi, and H. Chen, “Planar focusing antenna design by using coordinate transformation technology,” Appl. Phys. Lett. 91(25), 253509 (2007). [CrossRef]
27. S. Li, Y. Zhou, J. Dong, X. Zhang, E. Cassan, J. Hou, C. Yang, S. Chen, D. Gao, and H. Chen, “Universal multimode waveguide crossing based on transformation optics,” Optica 5(12), 1549–1556 (2018). [CrossRef]
28. K. Yao, X. Jiang, and H. Chen, “Collimating lenses from non-Euclidean transformation optics,” New J. Phys. 14(2), 023011 (2012). [CrossRef]
29. S. Zhang, C. Xia, and N. Fang, “Broadband Acoustic Cloak for Ultrasound Waves,” Phys. Rev. Lett. 106(2), 024301 (2011). [CrossRef]
30. S. H. Kim, Pham-Van Sy, and M. P. Das, “Acoustic Eaton lens array and its fluid application,” Mod. Phys. Lett. B 31(08), 1750081 (2017). [CrossRef]
31. L. Bai, H. Y. Dong, G. Y. Song, Q. Cheng, B. Huang, W. X. Jiang, and T. J. Cui, “Impedance-Matching Wavefront-Transformation Lens Based on Ascoustic Metamaterials,” Adv. Mater. Technol. 3(11), 1800064 (2018). [CrossRef]
32. A. Mikaelian and A. Prokhorov, “V self-focusing media with variable index of refraction,” Prog. Opt. 17, 279–345 (1980). [CrossRef]
33. H. Chen, “Imaging along conformal curves,” Phys. Rev. A 98(4), 043843 (2018). [CrossRef]
34. A. J. Makowski and K. J. Gorska, “Quantization of the Maxwell fish-eye problem and the quantum-classical correspondence,” Phys. Rev. A 79(5), 052116 (2009). [CrossRef]
35. S. Fl, Practical Quantum Mechanics (Springer Science & Business Media, 1999).
36. H. Chen and W. Xiao, “Morse lens,” Chin. Opt. Lett. 18(6), 062403 (2020). [CrossRef]
37. S.-H. Kim, “Retroreflector approximation of a generalized Eaton lens,” J Mod. Opt. 59(9), 839–842 (2012). [CrossRef]
38. L. Xu, X. Wang, T. Tyc, C. Sheng, S. Zhu, H. Liu, and H. Chen, “Light rays and waves on geodesic lenses,” Photonics Res. 7(11), 1266–1272 (2019). [CrossRef]
39. Y. Zhou, Z. Hao, P. Zhao, and H. Chen, “Solid Immersion Maxwell’s Fish-Eye Lens Without Drain,” Phys. Rev. Appl. 17(3), 034039 (2022). [CrossRef]
40. L. Xu and H. Chen, “Coherent perfect absorber makes a perfect drain for Maxwell’s fish-eye lens,” Europhys. Lett. 100(3), 34001 (2012). [CrossRef]
41. Q. Ba, Y. Zhou, J. Li, W. Xiao, L. Ye, Y. Liu, J. -h. Lhen, H. Chen, and H. Y. Chen, “Conformal optical black hole for cavity,” eLight 2(1), 19 (2022). [CrossRef]
42. T. Tyc and U. Leonhardt, “Transmutation of singularities in optical instruments,” New J. Phys. 10(11), 115038 (2008). [CrossRef]
43. M. Sarbort and T. Tyc, “Spherical media and geodesic lenses in geometrical optics,” J. Opt. 14(7), 075705 (2012). [CrossRef]
44. T. Zentgraf, Y. Liu, M. H. Mikkelsen, J. Valentine, and X. Zhang, “Plasmonic luneburg and eaton lenses,” Nat. Nanotechnol. 6(3), 151–155 (2011). [CrossRef]
45. Q. Chen, S. A. R. Horsley, N. J. G. Fonseca, T. Tyc, and O. Quevedo-Teruel, “Double-layer geodesic and gradient-index lenses,” Nat. Commun. 13(1), 2354 (2022). [CrossRef]
46. G. Du, M. Liang, R. A. Sabory-Garcia, C. Liu, and H. Xin, “3-D Printing Implementation of an X-band Eaton Lens for Beam Deflection,” Antennas Wirel. Propag. Lett. 15, 1487–1490 (2016). [CrossRef]
47. J. Y. Li and M. Ng Mou Kehn, “The 90° Rotating Eaton Lens Synthesized by Metasurfaces,” Antennas Wirel. Propag. Lett. 17(7), 1247–1251 (2018). [CrossRef]
48. Q. Wu, X. Feng, R. Chen, C. Gu, S. Li, H. Li, Y. Xu, Y. Lai, B. Hou, H. Chen, and others, “An inside-out Eaton lens made of H-fractal metamaterials,” Appl. Phys. Lett. 101(3), 031903 (2012). [CrossRef]
