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 [13] is an efficient tool to manipulate the propagation of light to design various optical devices, such as cloaks [46], field concentrators [7,8], rotators [9,10] and imaging devices [11,12]. Maxwell’s fisheye lens [13] is an absolute instrument [1418] 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 [2325] (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 [3335], 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]

$$n(r) = \frac{{2{r^{m - 1}}}}{{1 + {r^{2m}}}}$$
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.

 

Fig. 1. (a) Ray trajectory of GMFE lens mirror with parameter m = 1 (b) m = 1.5 and (c) m = 2. The yellow circles denote the sources in the lenses and the yellow pentagrams denote the images inside the lenses. (d) a schematic of such a system with parameter m = 2. The blue circles denote the CPAs located at the image positions. The blue curve on the right side denote the electric field distribution on the black dotted circle (where source and images locate).

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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 + 6.5412i 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 0.0197λ 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).

 

Fig. 2. (a) The electric field of Maxwell’s fisheye lens without a CPA and (c) with a CPA. (b) and (d) denote the distribution of electric field on the dotted source/image circle (see Fig. 1(d)) respectively. (e) The amplitude of electric field on the dotted source/image circle with a shifted CPA.

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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 7π/4 position and the corresponding image locates at 3π/4 position. We move the CPA from π/4 to 5π/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 3π/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 + 6.0331i 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.

 

Fig. 3. The electric field of GMFE lenses with parameter m = 1.5 (a) without CPA, (c) with one CPA and (e) with two CPAs. (b), (d) and (f) denote the distribution of electric field on the dotted source/image circle (see Fig. 1(d)) respectively.

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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 + 5.5106i 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 0.0216λ here), as shown in Fig. 4(d).

 

Fig. 4. The electric field of GMFE lenses with parameter m = 2 (a) without CPA, (c) with one CPA on the opposite of the source, (e) with one CPA close to the source, (g) with two CPAs that are symmetric about the source, (i) with two CPAs that are asymmetric about the source, and (k) with three CPAs. (b), (d), (f), (h), (j) and (l) denote the distribution of electric field on the dotted source/image circle (see Fig. 1(d)) respectively.

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Next, we embed the CPA at ($- \sqrt {2} $/4, $- \sqrt {2} $/4) that close to the source. It is no surprising that CPA enhances the resolution of image, as shown in Fig. 4(e), which is comparable to the GMFE lens with m = 1.5. We can verify it by contrasting with the amplitude of electric field on the source/image circle shown in Fig. 3(d) and Fig. 4(f), the imaging resolution shown in Fig. 4(f) (about 0.0207λ) is slightly smaller than that in Fig. 3(d) (about 0.0201λ), which can be attributed to the type of GMFE lens. Hence, it can be considered that CPA is an almost perfect drain for GMFE lens. To prove our assumption, further research that embed more CPAs into GMFE is needed. Firstly, we embed two CPAs into GMFE with m = 2 and two cases are included, one is embedded at ($- \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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References

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  1. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
    [Crossref]
  2. U. Leonhardt, “Optical conformal mapping,” Science 312(5781), 1777–1780 (2006).
    [Crossref]
  3. H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Mater. 9(5), 387–396 (2010).
    [Crossref]
  4. D. Schurig, J. Mock, B. Justice, S. A. Cummer, J. B. Pendry, A. Starr, and D. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
    [Crossref]
  5. J. Li and J. B. Pendry, “Hiding under the carpet: a new strategy for cloaking,” Phys. Rev. Lett. 101(20), 203901 (2008).
    [Crossref]
  6. R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science 323(5912), 366–369 (2009).
    [Crossref]
  7. M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell’s equations,” Photonics Nanostruct. 6(1), 87–95 (2008).
    [Crossref]
  8. M. M. Sadeghi, S. Li, L. Xu, B. Hou, and H. Y. Chen, “Transformation optics with Fabry-Perot resonances,” Sci. Rep. 5(1), 8680 (2015).
    [Crossref]
  9. H. Y. Chen and C. T. Chan, “Transformation media that rotate electromagnetic fields,” Appl. Phys. Lett. 90(24), 241105 (2007).
    [Crossref]
  10. H. Y. 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]
  11. D. A. Roberts, N. Kundtz, and D. R. Smith, “Optical lens compression via transformation optics,” Opt. Express 17(19), 16535–16542 (2009).
    [Crossref]
  12. N. Kundtz and D. R. Smith, “Extreme-angle broadband metamaterial lens,” Nat. Mater. 9(2), 129–132 (2010).
    [Crossref]
  13. J. C. Maxwell, “Solutions of Problems, problem no. 2,” Camb. Dublin Math. J. 8, 188–189 (1854).
  14. J. L. Synge, “The absolute optical instrument,” Trans. Amer. Math. Soc. 44(1), 32 (1938).
    [Crossref]
  15. J. C. Minano, “Perfect imaging in a homogeneous three-dimensional region,” Opt. Express 14(21), 9627–9635 (2006).
    [Crossref]
  16. 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]
  17. U. Leonhardt and T. Philbin, Geometry and Light: The Science of Invisibility (Dover, 2010).
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  48. S. Hadi Badri and M. M. Gilarlue, “Silicon nitride waveguide devices based on gradient-index lenses implemented by subwavelength silicon grating metamaterials,” Appl. Opt. 59(17), 5269–5275 (2020).
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2020 (2)

S. Hadi Badri, M. M. Gilarlue, and S. G. Gavgani, “Ultra-thin silicon-on-insulator waveguide bend based on truncated Eaton lens implemented by varying the guiding layer thickness,” Photonics Nanostruct. 39, 100766 (2020).
[Crossref]

S. Hadi Badri and M. M. Gilarlue, “Silicon nitride waveguide devices based on gradient-index lenses implemented by subwavelength silicon grating metamaterials,” Appl. Opt. 59(17), 5269–5275 (2020).
[Crossref]

2018 (4)

O. Bitton, R. Bruch, and U. Leonhardt, “Two-Dimensional Maxwell Fisheye for Integrated Optics,” Phys. Rev. Appl. 10(4), 044059 (2018).
[Crossref]

S. Li, Y. Zhou, J. Dong, X. Zhang, E. Cassan, J. Hou, C. Yang, S. Chen, D. Gao, and H. Y. Chen, “Universal multimode waveguide crossing based on transformation optics,” Optica 5(12), 1549–1556 (2018).
[Crossref]

H. Y. Chen, “Imaging along conformal curves,” Phys. Rev. A 98(4), 043843 (2018).
[Crossref]

Y. Fu, Y. Xu, H. Chen, and S. A. Cummer, “Coherent perfect absorption and laser modes in a cylindrical structure of conjugate metamaterials,” New J. Phys. 20(1), 013015 (2018).
[Crossref]

2017 (2)

H. A. Li, S. Suwunnarat, R. Fleischmann, H. Schanz, and T. Kottos, “Random Matrix Theory Approach to Chaotic Coherent Perfect Absorbers,” Phys. Rev. Lett. 118(4), 044101 (2017).
[Crossref]

D. G. Baranov, A. Krasnok, T. Shegai, A. Alu, and Y. D. Chong, “Coherent perfect absorbers: linear control of light with light,” Nat. Rev. Mater. 2(12), 17064 (2017).
[Crossref]

2015 (5)

L. Xu and H. Chen, “Conformal transformation optics,” Nat. Photonics 9(1), 15–23 (2015).
[Crossref]

M. M. Sadeghi, S. Li, L. Xu, B. Hou, and H. Y. Chen, “Transformation optics with Fabry-Perot resonances,” Sci. Rep. 5(1), 8680 (2015).
[Crossref]

A. J. Danner, H. L. Dao, and T. Tyc, “The Lissajous lens: a three-dimensional absolute optical instrument without spherical symmetry,” Opt. Express 23(5), 5716–5722 (2015).
[Crossref]

T. Roger, S. Vezzoli, E. Bolduc, J. Valente, J. J. F. Heitz, J. Jeffers, C. Soci, J. Leach, C. Couteau, N. I. Zheludev, and D. Faccio, “Coherent perfect absorption in deeply subwavelength films in the single-photon regime,” Nat. Commun. 6(1), 7031 (2015).
[Crossref]

U. Leonhardt and S. Sahebdivan, “Theory of Maxwell’s fish eye with mutually interacting sources and drains,” Phys. Rev. A 92(5), 053848 (2015).
[Crossref]

2014 (7)

J. C. Minano, J. Sanchez-Dehesa, J. C. Gonzalez, P. Benítez, D. Grabovivkic, J. Carbonell, and H. Ahmadpanahi, “Experimental evidence of super-resolution better than λ/105 with positive refraction,” New J. Phys. 16(3), 033015 (2014).
[Crossref]

T. Tyc and A. Danner, “Resolution of Maxwell's fisheye with an optimal active drain,” New J. Phys. 16(6), 063001 (2014).
[Crossref]

Y. Sun, W. Tan, H. Q. Li, J. S. Li, and H. Chen, “Experimental Demonstration of a Coherent Perfect Absorber with PT Phase Transition,” Phys. Rev. Lett. 112(14), 143903 (2014).
[Crossref]

P. J. Wei, C. Croenne, S. T. Chu, and J. S. Li, “Symmetrical and anti-symmetrical coherent perfect absorption for acoustic waves,” Appl. Phys. Lett. 104(12), 121902 (2014).
[Crossref]

J. Z. Song, P. Bai, Z. H. Hang, and Y. Lai, “Acoustic coherent perfect absorbers,” New J. Phys. 16(3), 033026 (2014).
[Crossref]

G. C. Ma, M. Yang, S. W. Xiao, Z. Y. Yang, and P. Sheng, “Acoustic metasurface with hybrid resonances,” Nat. Mater. 13(9), 873–878 (2014).
[Crossref]

S. Zanotto, F. P. Mezzapesa, F. Bianco, G. Biasiol, L. Baldacci, M. S. Vitiello, L. Sorba, R. Colombelli, and A. Tredicucci, “Perfect energy-feeding into strongly coupled systems and interferometric control of polariton absorption,” Nat. Phys. 10(11), 830–834 (2014).
[Crossref]

2013 (1)

2012 (4)

H. Noh, Y. Chong, A. D. Stone, and H. Cao, “Perfect coupling of light to surface plasmons by coherent absorption,” Phys. Rev. Lett. 108(18), 186805 (2012).
[Crossref]

J. F. Zhang, K. F. MacDonald, and N. I. Zheludev, “Controlling light-with-light without nonlinearity,” Light: Sci. Appl. 1(7), e18 (2012).
[Crossref]

J. W. Yoon, G. M. Koh, S. H. Song, and R. Magnusson, “Measurement and Modeling of a Complete Optical Absorption and Scattering by Coherent Surface Plasmon-Polariton Excitation Using a Silver Thin-Film Grating,” Phys. Rev. Lett. 109(25), 257402 (2012).
[Crossref]

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]

2011 (3)

J. C. González, P. Benítez, and J. C. Miñano, “Perfect drain for the Maxwell fish eye lens,” New J. Phys. 13(2), 023038 (2011).
[Crossref]

W. J. Wan, Y. D. Chong, L. Ge, H. Noh, A. D. Stone, and H. Cao, “Time-Reversed Lasing and Interferometric Control of Absorption,” Science 331(6019), 889–892 (2011).
[Crossref]

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]

2010 (4)

N. Kundtz and D. R. Smith, “Extreme-angle broadband metamaterial lens,” Nat. Mater. 9(2), 129–132 (2010).
[Crossref]

H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Mater. 9(5), 387–396 (2010).
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Y. D. Chong, L. Ge, H. Cao, and A. D. Stone, “Coherent perfect absorbers: time-reversed lasers,” Phys. Rev. Lett. 105(5), 053901 (2010).
[Crossref]

Q. Cheng, T. J. Cui, W. X. Jiang, and B. G. Cai, “An omnidirectional electromagnetic absorber made of metamaterials,” New J. Phys. 12(6), 063006 (2010).
[Crossref]

2009 (4)

U. Leonhardt, “Perfect imaging without negative refraction,” New J. Phys. 11(9), 093040 (2009).
[Crossref]

R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science 323(5912), 366–369 (2009).
[Crossref]

H. Y. 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]

D. A. Roberts, N. Kundtz, and D. R. Smith, “Optical lens compression via transformation optics,” Opt. Express 17(19), 16535–16542 (2009).
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2008 (2)

J. Li and J. B. Pendry, “Hiding under the carpet: a new strategy for cloaking,” Phys. Rev. Lett. 101(20), 203901 (2008).
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M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell’s equations,” Photonics Nanostruct. 6(1), 87–95 (2008).
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2007 (1)

H. Y. Chen and C. T. Chan, “Transformation media that rotate electromagnetic fields,” Appl. Phys. Lett. 90(24), 241105 (2007).
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2006 (4)

D. Schurig, J. Mock, B. Justice, S. A. Cummer, J. B. Pendry, A. Starr, and D. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
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J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
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U. Leonhardt, “Optical conformal mapping,” Science 312(5781), 1777–1780 (2006).
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J. C. Minano, “Perfect imaging in a homogeneous three-dimensional region,” Opt. Express 14(21), 9627–9635 (2006).
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1980 (1)

A. Mikaelian and A. Prokhorov, “V Self-Focusing Media with Variable Index of Refraction,” Prog. Opt. 17, 279–345 (1980).
[Crossref]

1971 (1)

Y. N. Demkov and V. N. Ostrovsky, “Internal symmetry of the Maxwell “fish-eye” problem and the Fock group for the Hydrogen atom,” Sov. Phys. JETP 33, 1083–1087 (1971).

1938 (1)

J. L. Synge, “The absolute optical instrument,” Trans. Amer. Math. Soc. 44(1), 32 (1938).
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1854 (1)

J. C. Maxwell, “Solutions of Problems, problem no. 2,” Camb. Dublin Math. J. 8, 188–189 (1854).

Ahmadpanahi, H.

J. C. Minano, J. Sanchez-Dehesa, J. C. Gonzalez, P. Benítez, D. Grabovivkic, J. Carbonell, and H. Ahmadpanahi, “Experimental evidence of super-resolution better than λ/105 with positive refraction,” New J. Phys. 16(3), 033015 (2014).
[Crossref]

Alu, A.

D. G. Baranov, A. Krasnok, T. Shegai, A. Alu, and Y. D. Chong, “Coherent perfect absorbers: linear control of light with light,” Nat. Rev. Mater. 2(12), 17064 (2017).
[Crossref]

Ao, X.

H. Y. 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]

Bai, P.

J. Z. Song, P. Bai, Z. H. Hang, and Y. Lai, “Acoustic coherent perfect absorbers,” New J. Phys. 16(3), 033026 (2014).
[Crossref]

Baldacci, L.

S. Zanotto, F. P. Mezzapesa, F. Bianco, G. Biasiol, L. Baldacci, M. S. Vitiello, L. Sorba, R. Colombelli, and A. Tredicucci, “Perfect energy-feeding into strongly coupled systems and interferometric control of polariton absorption,” Nat. Phys. 10(11), 830–834 (2014).
[Crossref]

Baranov, D. G.

D. G. Baranov, A. Krasnok, T. Shegai, A. Alu, and Y. D. Chong, “Coherent perfect absorbers: linear control of light with light,” Nat. Rev. Mater. 2(12), 17064 (2017).
[Crossref]

Benítez, P.

J. C. Minano, J. Sanchez-Dehesa, J. C. Gonzalez, P. Benítez, D. Grabovivkic, J. Carbonell, and H. Ahmadpanahi, “Experimental evidence of super-resolution better than λ/105 with positive refraction,” New J. Phys. 16(3), 033015 (2014).
[Crossref]

J. C. González, P. Benítez, and J. C. Miñano, “Perfect drain for the Maxwell fish eye lens,” New J. Phys. 13(2), 023038 (2011).
[Crossref]

Bering, K.

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]

Bianco, F.

S. Zanotto, F. P. Mezzapesa, F. Bianco, G. Biasiol, L. Baldacci, M. S. Vitiello, L. Sorba, R. Colombelli, and A. Tredicucci, “Perfect energy-feeding into strongly coupled systems and interferometric control of polariton absorption,” Nat. Phys. 10(11), 830–834 (2014).
[Crossref]

Biasiol, G.

S. Zanotto, F. P. Mezzapesa, F. Bianco, G. Biasiol, L. Baldacci, M. S. Vitiello, L. Sorba, R. Colombelli, and A. Tredicucci, “Perfect energy-feeding into strongly coupled systems and interferometric control of polariton absorption,” Nat. Phys. 10(11), 830–834 (2014).
[Crossref]

Bitton, O.

O. Bitton, R. Bruch, and U. Leonhardt, “Two-Dimensional Maxwell Fisheye for Integrated Optics,” Phys. Rev. Appl. 10(4), 044059 (2018).
[Crossref]

Bolduc, E.

T. Roger, S. Vezzoli, E. Bolduc, J. Valente, J. J. F. Heitz, J. Jeffers, C. Soci, J. Leach, C. Couteau, N. I. Zheludev, and D. Faccio, “Coherent perfect absorption in deeply subwavelength films in the single-photon regime,” Nat. Commun. 6(1), 7031 (2015).
[Crossref]

Bruch, R.

O. Bitton, R. Bruch, and U. Leonhardt, “Two-Dimensional Maxwell Fisheye for Integrated Optics,” Phys. Rev. Appl. 10(4), 044059 (2018).
[Crossref]

Cai, B. G.

Q. Cheng, T. J. Cui, W. X. Jiang, and B. G. Cai, “An omnidirectional electromagnetic absorber made of metamaterials,” New J. Phys. 12(6), 063006 (2010).
[Crossref]

Cao, H.

H. Noh, Y. Chong, A. D. Stone, and H. Cao, “Perfect coupling of light to surface plasmons by coherent absorption,” Phys. Rev. Lett. 108(18), 186805 (2012).
[Crossref]

W. J. Wan, Y. D. Chong, L. Ge, H. Noh, A. D. Stone, and H. Cao, “Time-Reversed Lasing and Interferometric Control of Absorption,” Science 331(6019), 889–892 (2011).
[Crossref]

Y. D. Chong, L. Ge, H. Cao, and A. D. Stone, “Coherent perfect absorbers: time-reversed lasers,” Phys. Rev. Lett. 105(5), 053901 (2010).
[Crossref]

Carbonell, J.

J. C. Minano, J. Sanchez-Dehesa, J. C. Gonzalez, P. Benítez, D. Grabovivkic, J. Carbonell, and H. Ahmadpanahi, “Experimental evidence of super-resolution better than λ/105 with positive refraction,” New J. Phys. 16(3), 033015 (2014).
[Crossref]

Cassan, E.

Chan, C. T.

H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Mater. 9(5), 387–396 (2010).
[Crossref]

H. Y. 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]

H. Y. Chen and C. T. Chan, “Transformation media that rotate electromagnetic fields,” Appl. Phys. Lett. 90(24), 241105 (2007).
[Crossref]

Chen, H.

Y. Fu, Y. Xu, H. Chen, and S. A. Cummer, “Coherent perfect absorption and laser modes in a cylindrical structure of conjugate metamaterials,” New J. Phys. 20(1), 013015 (2018).
[Crossref]

L. Xu and H. Chen, “Conformal transformation optics,” Nat. Photonics 9(1), 15–23 (2015).
[Crossref]

Y. Sun, W. Tan, H. Q. Li, J. S. Li, and H. Chen, “Experimental Demonstration of a Coherent Perfect Absorber with PT Phase Transition,” Phys. Rev. Lett. 112(14), 143903 (2014).
[Crossref]

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]

H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Mater. 9(5), 387–396 (2010).
[Crossref]

Chen, H. Y.

S. Li, Y. Zhou, J. Dong, X. Zhang, E. Cassan, J. Hou, C. Yang, S. Chen, D. Gao, and H. Y. Chen, “Universal multimode waveguide crossing based on transformation optics,” Optica 5(12), 1549–1556 (2018).
[Crossref]

H. Y. Chen, “Imaging along conformal curves,” Phys. Rev. A 98(4), 043843 (2018).
[Crossref]

M. M. Sadeghi, S. Li, L. Xu, B. Hou, and H. Y. Chen, “Transformation optics with Fabry-Perot resonances,” Sci. Rep. 5(1), 8680 (2015).
[Crossref]

H. Y. 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]

H. Y. Chen and C. T. Chan, “Transformation media that rotate electromagnetic fields,” Appl. Phys. Lett. 90(24), 241105 (2007).
[Crossref]

Chen, S.

S. Li, Y. Zhou, J. Dong, X. Zhang, E. Cassan, J. Hou, C. Yang, S. Chen, D. Gao, and H. Y. Chen, “Universal multimode waveguide crossing based on transformation optics,” Optica 5(12), 1549–1556 (2018).
[Crossref]

H. Y. 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]

Cheng, Q.

Q. Cheng, T. J. Cui, W. X. Jiang, and B. G. Cai, “An omnidirectional electromagnetic absorber made of metamaterials,” New J. Phys. 12(6), 063006 (2010).
[Crossref]

Chin, J. Y.

R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science 323(5912), 366–369 (2009).
[Crossref]

Chong, Y.

H. Noh, Y. Chong, A. D. Stone, and H. Cao, “Perfect coupling of light to surface plasmons by coherent absorption,” Phys. Rev. Lett. 108(18), 186805 (2012).
[Crossref]

Chong, Y. D.

D. G. Baranov, A. Krasnok, T. Shegai, A. Alu, and Y. D. Chong, “Coherent perfect absorbers: linear control of light with light,” Nat. Rev. Mater. 2(12), 17064 (2017).
[Crossref]

N. Gutman, A. A. Sukhorukov, Y. D. Chong, and C. M. de Sterke, “Coherent perfect absorption and reflection in slow-light waveguides,” Opt. Lett. 38(23), 4970–4973 (2013).
[Crossref]

W. J. Wan, Y. D. Chong, L. Ge, H. Noh, A. D. Stone, and H. Cao, “Time-Reversed Lasing and Interferometric Control of Absorption,” Science 331(6019), 889–892 (2011).
[Crossref]

Y. D. Chong, L. Ge, H. Cao, and A. D. Stone, “Coherent perfect absorbers: time-reversed lasers,” Phys. Rev. Lett. 105(5), 053901 (2010).
[Crossref]

Chu, S. T.

P. J. Wei, C. Croenne, S. T. Chu, and J. S. Li, “Symmetrical and anti-symmetrical coherent perfect absorption for acoustic waves,” Appl. Phys. Lett. 104(12), 121902 (2014).
[Crossref]

Colombelli, R.

S. Zanotto, F. P. Mezzapesa, F. Bianco, G. Biasiol, L. Baldacci, M. S. Vitiello, L. Sorba, R. Colombelli, and A. Tredicucci, “Perfect energy-feeding into strongly coupled systems and interferometric control of polariton absorption,” Nat. Phys. 10(11), 830–834 (2014).
[Crossref]

Couteau, C.

T. Roger, S. Vezzoli, E. Bolduc, J. Valente, J. J. F. Heitz, J. Jeffers, C. Soci, J. Leach, C. Couteau, N. I. Zheludev, and D. Faccio, “Coherent perfect absorption in deeply subwavelength films in the single-photon regime,” Nat. Commun. 6(1), 7031 (2015).
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Q. Cheng, T. J. Cui, W. X. Jiang, and B. G. Cai, “An omnidirectional electromagnetic absorber made of metamaterials,” New J. Phys. 12(6), 063006 (2010).
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Y. Sun, W. Tan, H. Q. Li, J. S. Li, and H. Chen, “Experimental Demonstration of a Coherent Perfect Absorber with PT Phase Transition,” Phys. Rev. Lett. 112(14), 143903 (2014).
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U. Leonhardt and S. Sahebdivan, “Theory of Maxwell’s fish eye with mutually interacting sources and drains,” Phys. Rev. A 92(5), 053848 (2015).
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J. C. Minano, J. Sanchez-Dehesa, J. C. Gonzalez, P. Benítez, D. Grabovivkic, J. Carbonell, and H. Ahmadpanahi, “Experimental evidence of super-resolution better than λ/105 with positive refraction,” New J. Phys. 16(3), 033015 (2014).
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H. A. Li, S. Suwunnarat, R. Fleischmann, H. Schanz, and T. Kottos, “Random Matrix Theory Approach to Chaotic Coherent Perfect Absorbers,” Phys. Rev. Lett. 118(4), 044101 (2017).
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D. Schurig, J. Mock, B. Justice, S. A. Cummer, J. B. Pendry, A. Starr, and D. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
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N. Kundtz and D. R. Smith, “Extreme-angle broadband metamaterial lens,” Nat. Mater. 9(2), 129–132 (2010).
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D. A. Roberts, N. Kundtz, and D. R. Smith, “Optical lens compression via transformation optics,” Opt. Express 17(19), 16535–16542 (2009).
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M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell’s equations,” Photonics Nanostruct. 6(1), 87–95 (2008).
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J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
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T. Roger, S. Vezzoli, E. Bolduc, J. Valente, J. J. F. Heitz, J. Jeffers, C. Soci, J. Leach, C. Couteau, N. I. Zheludev, and D. Faccio, “Coherent perfect absorption in deeply subwavelength films in the single-photon regime,” Nat. Commun. 6(1), 7031 (2015).
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J. Z. Song, P. Bai, Z. H. Hang, and Y. Lai, “Acoustic coherent perfect absorbers,” New J. Phys. 16(3), 033026 (2014).
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J. W. Yoon, G. M. Koh, S. H. Song, and R. Magnusson, “Measurement and Modeling of a Complete Optical Absorption and Scattering by Coherent Surface Plasmon-Polariton Excitation Using a Silver Thin-Film Grating,” Phys. Rev. Lett. 109(25), 257402 (2012).
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D. Schurig, J. Mock, B. Justice, S. A. Cummer, J. B. Pendry, A. Starr, and D. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
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H. Noh, Y. Chong, A. D. Stone, and H. Cao, “Perfect coupling of light to surface plasmons by coherent absorption,” Phys. Rev. Lett. 108(18), 186805 (2012).
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Y. Sun, W. Tan, H. Q. Li, J. S. Li, and H. Chen, “Experimental Demonstration of a Coherent Perfect Absorber with PT Phase Transition,” Phys. Rev. Lett. 112(14), 143903 (2014).
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H. A. Li, S. Suwunnarat, R. Fleischmann, H. Schanz, and T. Kottos, “Random Matrix Theory Approach to Chaotic Coherent Perfect Absorbers,” Phys. Rev. Lett. 118(4), 044101 (2017).
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T. Roger, S. Vezzoli, E. Bolduc, J. Valente, J. J. F. Heitz, J. Jeffers, C. Soci, J. Leach, C. Couteau, N. I. Zheludev, and D. Faccio, “Coherent perfect absorption in deeply subwavelength films in the single-photon regime,” Nat. Commun. 6(1), 7031 (2015).
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H. Y. 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).
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Figures (4)

Fig. 1.
Fig. 1. (a) Ray trajectory of GMFE lens mirror with parameter m = 1 (b) m = 1.5 and (c) m = 2. The yellow circles denote the sources in the lenses and the yellow pentagrams denote the images inside the lenses. (d) a schematic of such a system with parameter m = 2. The blue circles denote the CPAs located at the image positions. The blue curve on the right side denote the electric field distribution on the black dotted circle (where source and images locate).
Fig. 2.
Fig. 2. (a) The electric field of Maxwell’s fisheye lens without a CPA and (c) with a CPA. (b) and (d) denote the distribution of electric field on the dotted source/image circle (see Fig. 1(d)) respectively. (e) The amplitude of electric field on the dotted source/image circle with a shifted CPA.
Fig. 3.
Fig. 3. The electric field of GMFE lenses with parameter m = 1.5 (a) without CPA, (c) with one CPA and (e) with two CPAs. (b), (d) and (f) denote the distribution of electric field on the dotted source/image circle (see Fig. 1(d)) respectively.
Fig. 4.
Fig. 4. The electric field of GMFE lenses with parameter m = 2 (a) without CPA, (c) with one CPA on the opposite of the source, (e) with one CPA close to the source, (g) with two CPAs that are symmetric about the source, (i) with two CPAs that are asymmetric about the source, and (k) with three CPAs. (b), (d), (f), (h), (j) and (l) denote the distribution of electric field on the dotted source/image circle (see Fig. 1(d)) respectively.

Equations (1)

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n ( r ) = 2 r m 1 1 + r 2 m

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