Transformation Optics: From Classic Theory and Applications to its New Branches

Transcription

1 REVIEW ARTICLE Transformation Optics: From Classic Theory and Applications to its New Branches Fei Sun, Bin Zheng, Hongsheng Chen, Wei Jiang, Shuwei Guo, Yichao Liu, Yungui Ma, and Sailing He* In the modern world, the ability to manipulate and control electromagnetic waves has greatly changed people s lives. Novel optical and electromagnetic phenomena and devices will lead to new scientific trends and techniques in the future. The exploration of new theories of optical design and new materials for optical engineering has attracted great attention in recent years. Transformation optics (TO) provides a new way to design optical devices with extraordinary predesigned functions such as invisibility cloaks and electromagnetic wormholes. As the development of artificial electromagnetic media (e.g. metamaterials and metasurfaces) progresses, many of these novel optical devices designed by TO have been experimentally demonstrated and used in specific applications. Starting from the basic theory of transformation optics, we review its applications, extensions, new branches and recent developments in this paper. 1. Introduction In the first part of this review, we begin from the fundamental principles of TO and show the basic formulas to design transformation media. Then, we introduce classic applications of TO. F. Sun, W. Jiang, S. Guo, Y. Liu, Y. Ma, S. He Centre for Optical and Electromagnetic Research Zhejiang Provincial Key Laboratory for Sensing Technologies JORCEP East Building #5 Zhejiang University Hangzhou , P. R. China sailing@kth.se B. Zheng, H. Chen College of Information Science & Electronic Engineering Zhejiang University Hangzhou , P. R. China B. Zheng, H. Chen State Key Laboratory of Modern Optical Instrumentation Zhejiang University Hangzhou , P. R. China B. Zheng, H. Chen The Electromagnetics Academy at Zhejiang University Zhejiang University Hangzhou , P. R. China S. He Department of Electromagnetic Engineering School of Electrical Engineering Royal Institute of Technology (KTH) S Stockholm, Sweden DOI: /lpor Background and Origin of TO Among these devices, we will focus on the most exciting application of TO, the invisibility cloak, and review recent progress on the experimental realization of optical cloaking (e.g. from a carpet optical cloak with a small concealed region to a natural light cloak for living creatures). In the second part of this review, we will introduce two extensions of TO: optical surface transformation (OST) and optic-null medium (ONM). Based on OST and ONM, many novel optical devices can be designed without cumbersome mathematical calculations. The entire design process consists of choosing suitable shapes of the input and output surfaces and the directions of the ONMs main axes. Optical media properties such as refractive index, permittivity and permeability and Maxwell s equations (or Fermat s principle in a geometrical-optic approximation) [1 3] determine the propagation of light. However, to design an optical device with a predesigned function, we need to determine the path of the light in advance and then determine which medium can achieve such a function (i.e. the light follows the predesigned path in this medium). Designing an optical device with a predesigned function is an electromagnetic inverse scattering problem, which is intractable using conventional optical design methods. Many classic optical devices designed using geometrical optics (i.e. based on Fermat s principle), [4] including Maxwell s fish-eye lens, [5,6] the Luneburg lens, [7 10] the Eaton lens, [11 13] Mikaelian s lens, [14 17] rely on inhomogeneous refractive indices to control the light s path. TO originates from a similar idea the medium permittivity and permeability represents a curved spatial geometry for the light. [18 20] Such an effective curved geometry in real space can be obtained by a coordinate transformation from a flat virtual space, often referred as the reference space. [18 25] As shown in Figure 1, we have two spaces in TO: the real space filled by the transformation medium where the light propagates in a predesigned curved path (Figure 1b), and the reference space where the light path is a straight line (Figure 1a). TO uses the material interpretation, [20] which holds that the effect of the curved (1 of 27)

2 Fei Sun is currently a post-doctor in College of Optical Science and Engineering, Zhejiang University, China. He received the Ph.D. degrees in electric engineering from the Royal Institute of Technology (KTH), Stockholm, Sweden in 2014, and in optical engineering from Zhejiang University, Hangzhou, China in 2015, respectively. Dr. Sun s current research interests include transformation optics, novel optical devices, electromagnetic metamaterials and acoustic devices. Bin Zheng is lecturer at the College of Information Science and Electronic Engineering in Zhejiang University (China) since He has been a Postdoctoral Researcher at Zhejiang University since He received the B.S. degree from Ningbo University (China) in 2010 and the Ph.D. degree from Zhejiang University (China) in His research interests include transformation optics, metamaterials, metasurfaces and invisibility cloaks. Hongsheng Chen is a Chang Jiang Scholar Distinguished Professor in the Electromagnetics Academy at Zhejiang University, China. He has been a Visiting Scientist (during ) and a Visiting Professor (during ) with the Research Laboratory of Electronics at Massachusetts Institute of Technology, USA. His current research interests include the areas of metamaterials, antennas, invisibility cloaking, transformation optics, and graphene. He is the coauthor of more than 180 international refereed journal papers. He serves on the Topical Editor of the Journal of Optics, the Editorial Board of the Nature s Scientific Reports, and Progress in Electromagnetics Research. Dr. Chen received the National Excellent Doctoral Dissertation Award in China (2008), the National Youth Top-notch Talent Support Program in China (2012), and the National Science Foundation for Distinguished Young Scholars of China (2016). His research work on invisibility cloak was selected in Science Development Report as one of the representative achievements of Chinese Scientists in Jiang Wei received the B.E. degree from Zhejiang University, Hangzhou, China in 2013, and is currently working toward the Ph.D. degree at Zhejiang University, Hangzhou, China. His main research interests include the design and application of transformation optics, metasurface, plasmonics and nano-antenna. Shuwei Guo has been a graduate student studying optical engineerng in Zhejiang University, China science He obtained his bachelor of engineering degree in His research interests are primarily concerned with transformation optics, sub-wavelength imaging and metamaterials. Yichao Liu is a member of Meta group of COER (Centre for Optical and Electromagnetic Research). He received his PhD from Zhejiang University in 2016 for his research on transformation optics and meta materials. Now he is a postdoctoral fellow of the COER. Yungui Ma is a professor of Zhejiang University in Faculty of Optical Science and Engineering. He received his Bachelor (physics) and PhD (physics) degrees from Lanzhou University (China) in 2000 and 2005, respectively. He worked in National University of Singapore (Singapore) as research fellow and later research scientist from 2005 to He joined Zhejiang University (China) in His current research includes metamaterials, nanophotonics, plasmonics and near-field heat transfer. In 2011, he was selected into the youth talent supporting program of Chinese Ministry of Education. Sailing He received the Licentiate of Technology and the Ph.D. degree in electromagnetic theory from the Royal Institute of Technology (KTH), Stockholm, Sweden, in 1991 and 1992, respectively. Since then he has worked at the same division of the Royal Institute of Technology as an assistant professor, an associate professor, and a full professor. Prof. He also serves as director for JORCEP (a Sino-Swedish joint research center of photonics) at Zhejiang University (China). Prof. He s current research interests include electromagnetic metamaterials, optoelectronics, optical sensing, applied electromagnetics, and biomedical applications. He has first-authored one monograph (Oxford University Press) and authored/coauthored about 500 papers in refereed international journals. He is a fellow of IEEE, OSA and SPIE (2 of 27)

3 Figure 1. The relation between the reference space and real space in TO (a) The reference space is a flat space in a Cartesian coordinate system. There is no electromagnetic media filled in this space (i.e. it is free space). Light propagates in a straight path (e.g. the red arrow). (b) The real space: after the coordinate transformation, the coordinate grid is curved (i.e. a curved geometry). geometry on the light is the same as the transformation medium s effect of the light. In the materials interpretation, the coordinates are still the Cartesian coordinate system in the real space, but the transformation medium (colored green in Figure 1b) is filling in the whole real space. Light propagates in a predesigned curved path in real space (red arrow in Figure 1b). This curved path can be designed by a coordinate transformation method in advance, and then the required transformation medium can be calculated by TO. Fundamental theories of TO can be found in Refs. [19] and [20]. In recent years, TO has become a powerful tool to control electromagnetic waves. [23 25] Many novel optical devices that cannot be designed by any previous methods (e.g. geometrical optics, wave diffusion theory, Fourier optics, etc.) have been designed and even experimentally demonstrated with TO, including an invisibility cloak, [20,26 32] wave concentrators, [33,34] rotators, [35 38] beam splitters, [39,40] compressors, [41] perfect lenses, [42 44] illusion optical devices, [45,46] PEC reshapers, [47,48] superscatterers, [49 52] optical black holes, [53 55] optical wormholes, [56,57] wavefront modulators, [58,59] waveguide devices, [60 62] and antenna radiation controllers, [63 65] etc. Among these works, there are some notable experimental milestones, such as the realizations of an invisibility cloak in the microwave band, [27,28] infrared band [29] and visiblelight band, [32] electromagnetic wave rotators for microwave [36] and terahertz waves, [37] and a microwave highly directive emitter. [63] The theoretical basis of TO is the form-invariance of Maxwell s equations under coordinate transformations, [66,67] which can also be extended to other physical fields due to the forminvariance of other kinds of physical equations (e.g. transformation acoustics, [68,69] transformation thermal dynamics, [70 72] etc.). Many novel devices in other physical fields have also been designed in a similar coordinate transformation method, such as acoustic cloaks [73 75], acoustic hyperlenses [76], thermal cloaks, [77 80] thermal illusions, [81 83] and temperature control devices, [84,85] etc. Some of these devices have been experimentally demonstrated, including acoustic hyperlenses for superresolution imaging, [76] thermal cloaks for heat protection, [77 80] thermal plates for uniform heating, [84] and some thermal illusion devices. [81,82] 1.2. Basic Theory of TO We assume that a medium of relative permittivity ε and permeability μ is filled in the reference space in Cartesian coordinates (x, y, z) in Figure 1a, and a medium of relative permittivity ε and permeability μ is filled in the real space in Cartesian coordinates (x, y, z ) in Figure 1b. TO illustrates how to make a connection between the media and fields in the two spaces by a coordinate transformation between these two spaces. Note that throughout the paper quantities with and without primes are in the real and reference spaces, respectively. The Jacobian matrix of the coordinate transformation is defined as: x x x x y z y y y A =. (1) x y z z x z y z z Under the requirements of the form-invariance of Maxwell s equations, the relation of the electromagnetic media between the real and reference spaces is given by TO: [20] Aε ε AT = det( A). (2) μ = AμAT det( A) (3 of 27)

4 Note that all other quantities appearing in Maxwell s equations are also transformed, which can be calculated by: [86,87] E = (A T ) 1 E H = (A T ) 1 H D = 1 det(a) AD B = 1 det(a) AB. (3) J = 1 det(a) AJ ρ = 1 det(a) ρ Equation (3) means that, if the fields and sources distribution in the reference space are known, we can also use the coordinate transformation to determine the corresponding fields and sources distribution in the real space. If the coordinate systems in the real and reference spaces are not Cartesian coordinate systems, but instead some other orthogonal coordinate systems, the Jacobian transformation matrix should be modified by: [88,89] h 1 x 1 h 1 x 1 h 1 x 1 h 1 x 1 h 2 x 2 h 3 x 3 A = h 2 x 2 h 2 x 2 h 2 x 2, (4) h 1 x 1 h 2 x 2 h 3 x 3 h 3 x 3 h 3 x 3 h 3 x 3 h 1 x 1 h 2 x 2 h 3 x 3 where h i = (i = 1, 2, 3) are the scale factors of the variables x i in the orthogonal coordinate system. For example, we have h r = h r = 1, h θ = h θ = r, h z = h z = 1 in the cylindrical coordinate system Optical Conformal Mapping In a geometrical-optics approximation (λ 0),wecanusethe Helmholtz equation to describe the propagation of light instead of Maxwell s equations. The Helmholtz equation is also form-invariant under optical conformal mapping (OCM). [90 92] The theory of 2D OCM can be derived from the general TO theory (Eq. (2)). [93] We consider two 2D planes with complex coordinates z = x + iy with refractive index n(x, y) and with complex coordinates z = x + iy with refractive index n (x, y ) in the reference and real spaces, respectively, for TEpolarized light (the electric field is orthogonal to the 2D plane). Assuming the coordinate transformation between 2D planes is a conformal mapping, the Cauchy Riemann condition is satisfied: x x = y y x. (5) y = y x In this case, the Jacobian transformation matrix in Eq. (1) can be simplified as: x x x y 0 A = x x y x 0. (6) Assuming the medium in the reference space is a nonmagnetic isotropic medium, μ = 1andε = n 2, and combining Eqs. (2) and (6), we obtain: ( ε = diag 1, 1, ( μ = diag 1, 1, ( x x ( x x 1 ) 2+ ( x y 1 ) 2+ ( x y ) 2 ) ε ) 2 ). (7) For a TE-polarized beam, only μ within the 2D plane and ε outside the plane determine the light path, and hence Eq. (7) can be reduced to: n = n d z d z,μ = 1. (8) Equation (8) is consistent with the formula of OCM derived from the form-invariance of the Helmholtz equation or Fermat s principle. [90 93] The most important feature of OCM is that the transformation medium (see Eq. (8)) is isotropic and nonmagnetic, which is much simpler than the general transformation medium given by TO. Many novel optical devices based on OCM have been designed, as it is easy to fabricate. [94 99] However, there are still two main limitations of OCM: the devices designed using OCM are 2D devices (we can achieve a 2.5D device with a rotation method [100] ), and the shape of the device cannot be arbitrary, as only two regions having the same conformal module can be linked by a conformal mapping. This requirement can be reduced by introducing a quasiconformal mapping with a negligibly small anisotropy [101] Extension of TO to DC Fields Since DC magnetic fields and DC electric fields also follow Maxwell s equations, the theory of TO (Eqs. (2) and (3)) can be directly utilized to design media to control the DC field. DC magnetic/electric cloaks [ ], DC magnetic/electric field concentrators, [ ] DC magnetic hoses, [109,110] and DC magnetic wormholes [111] have all been designed using a coordinate transformation method. We have also extended the theory of TO to magnetostatics with active magnets in the reference space. [112] This means that we can control the DC magnetic field s distribution produced by some magnets in a predesigned manner. For DC magnetic fields, the Maxwell s equation in the reference space reduces to the following form: { B = 0 (9) H = J (4 of 27)

5 Figure 2. Basic types of DC magnetic field illusions by transformed magnets and transformation medium. (a) Scaling magnets: a small transformed magnet wrapped by a transformation medium (colored green) in free space (see left subfigure) produces the same magnetic field outside the dashed boundary as a large magnet in free space (see right subfigure). (b) Overlapping magnets: A magnet in free space and a magnet in a transformation medium, each with strength B r can be equivalent to a magnet with strength 2B r located at the point of the free space magnet. (c) Cancelling magnets: A magnet in free space and an antimagnet in a transformation medium can be arranged to produce no magnetic field outside of the immediate neighborhood. (d) Real sample to achieve a strong DC magnetic field enhancement (Reproduced with permission from Ref. [113]. Copyright (2014) Prog. Electromagn. Res.). The relation between the quantities in the reference and real spaces are still given by Eqs. (2) and (3). If there are magnets in the reference space, the constitutive relation should be modified by: [112] { B = μ0 μh + B r (10) B = μ 0 μ H + B r where B r is the residual magnetic flux density of the magnet in the reference space. The residual magnetic flux density of the magnet in the real space can be determined by: [112] B r = 1 det(a) AB r. (11) Using TO for a DC magnetic field case (e.g. Eqs. (2), (3) and (11)), we can create many novel static magnetic field illusions produced by transformed magnets, including scaling magnets, cancelling magnets, and overlapping magnets (see Figure 2 and Ref. [112] for details). We have also experimentally demonstrated a DC magnetic concentrator designed by TO that can achieve a nearly uniform enhancement by a factor of 4.74 with cooling and 3.84 at room temperature in a relatively large free-space region (see Figure 2d). [113] Another Theoretical Extension of TO In the present review, we only focus on the spatial coordinate transformation (i.e. the time coordinate does not change). For a time-related coordinate transformation, Maxwell s equations are still form-invariant [19,22]. In this case, the transformation medium is more complex (e.g. it may involve a time-varying medium, an electric magnetic coupling medium, etc.). We can obtain more innovative devices by introducing time-related coordinate transformations, including space time cloaks, [ ] event carpets, [117] time traveling, [118] etc. [119] The theory of TO has also been further developed and refined in recent years. The basic TO theory we introduced above can only design impedance-matched media (i.e. the curved geometry s influence on both permittivity and permeability are the same; see Eq. (2)). There are many new branches in TO theory, in which we can design a much wider range of media, such as chiral media, [120,121], nonimpedance-matched media by a triple (5 of 27)

6 space time TO, [122] and bianisotropic media by a fieldtransformation method. [123,124] 1.3. Background of Artificial Materials Metamaterials are artificial materials, which are composed of subwavelength man-made units with special properties that cannot be found in nature such as a negative refractive index, zero refractive index, magnetic response in high-frequency bands, etc. [125,126] Generally, the electromagnetic response of a natural material is determined by the interaction between the atoms (or molecules) and electromagnetic waves. Since the sizes of atoms are several orders smaller than the wavelength of the electromagnetic wave, the natural materials response to the electromagnetic waves is the average effect of the countless atoms. Similarly, a metamaterial s response to electromagnetic waves can also be treated as the average effect of the artificial subwavelength units, so an effective medium theory is appropriate, as the sizes of the man-made units in metamaterials are still much smaller than the wavelength of the incident waves. With specific design of the artificial units, an extraordinary control over electromagnetic waves can be achieved and many novel TO-based optical devices realized with metamaterials. [10,26 28] The development of metamaterials originates from the implementation of a double negative index medium, which was theoretically proposed by Veslago in 1968 [127] and experimentally demonstrated by Smith in [128] Initially, various metamaterials structures were designed to demonstrate the negative refraction phenomenon in different frequency bands. Later, various metamaterial structures based on effective electromagnetic parameters were designed to demonstrate the performance of novel devices based on TO. Using effective medium theory, we can define the local effective permittivity and permeability of metamaterials and produce materials with desirable properties, such as high anisotropy, [11] gradual control, [28,129] polarization control in the visible-light regime, [130] and light absorption [131] (Figure 3). In recent years, the concept of a planar metamaterial (metasurface), consisting of single-layer or few-layer stacks of planar structures provided an alternative way to realize similar extraordinary electromagnetic properties. Metasurfaces have been utilized to realize various optical devices, including TO-based devices (more details can be found in recent reviews on metasurfaces [132,133] ). 2. Invisibility Cloaks With the help of TO, the fanciful concept of invisibility became a scientific possibility, [20] which triumphantly led to experimental verification. Due to the complex constitutive parameters required for the cloak, most of the early experiments were mainly in the microwave region. With the development of TO as well as metamaterials in the optical band, various invisibility cloaks for optical frequencies have been realized in recent years Background and Classification of Invisibility Cloaks There are many ways to design an invisibility cloak by TO, which can be classified into four broad types. The first is an optical Figure 3. Various designs and applications of metamaterials. (a) Invisibility cloak in the microwave band (Reproduced with permission from Ref. [28]. Copyright (2006) Science). (b) 3D gradient control to realize the Luneburg lens at optical frequencies (Reproduced with permission from Ref. [129]. Copyright (2016) Laser Photon. Rev.). (c) Near-field subwavelength focusing (Reproduced with permission from Ref. [130]. Copyright (2009) Phys. Rev. A). (d) Deformable broadband absorber (Reproduced with permission from Ref. [131]. Copyright (2014) Laser Photon. Rev.). isolation cloak, in which we have a concealed region that is totally optically isolated from the outside world. When an incoming electromagnetic wave enters an optical isolation cloak, it will be smoothly guided around the concealed region and redirected back to its original direction (i.e. no scattering happens). For example, we can simply use a point-extended transformation to design such cloak in a spherical coordinate system: [20,21] a + r b a, a r b r = b r, r > b. (12) θ = θ ϕ = ϕ where a and b are the inner and outer boundaries of the spherical cloak. As we can see from the above transformation, the center point r = 0 in the reference space is extended to a spherical region with radius r = a in the real space (the region within this inner boundary of the cloak is optically isolated from the outer space, and hence, it cannot be detected). The whole coordinate transformation is continuous. For the region outside the sphere r = b, we have the identity coordinate transformation (the medium is still free space outside the cloak), and hence, the cloak does not produce any influence on the electromagnetic wave when it passes through the device. With the help of TO (Eqs. (2) and (4)), (6 of 27)

7 Figure 4. Four different kinds of invisibility cloak. (a) The optical isolation cloak: the light beam is smoothly guided around the concealed region (i.e. not touching the concealed objects, and hence no scattering features of concealed objects are captured), and then redirected to its original direction (Reproduced with permission from Ref. [20]. Copyright (2006) Science). (b) The scattering cancellation cloak: the concealed object (e.g. the green star) is set near the cloak (e.g. the red star and the yellow shell). The red star in the yellow shell can be treated as the complementary medium of the green star from the perspective of TO. The whole system (i.e. the concealed object and the cloak) produces no scattering of the detecting wave. Note that the detecting wave touches the concealed object in this case. The scattering produced by the concealed object is cancelled by the scattering produced by the cloak. (c) The scattering cover-up cloak: We have a background object (e.g. a tree) which produces strong scattering. The concealed object (e.g. the blue star) is set inside the cloak (e.g. the pink circle) around the background object. For an outside observer, the scattering of the concealed object is overlapped by the background object of strong scattering, he/she can only see the tree without the star. (d) The carpet cloak (Reproduced with permission from Ref. [101]. Copyright (2008) Phys. Rev. Lett.). The concealed object is hidden under a curved patch of ground covered by the carpet cloak in the real space (left subfigure). What the outside viewer sees is a flat ground, without seeing that the concealed object is under it (right subfigure). we can obtain the required electromagnetic medium in region a<r <b to achieve such a cloak: ε r = μ r = b ( ) r a 2 b a r ε θ = μ θ = b. (13) b a ε ϕ = μ ϕ = b b a Figure 4a shows the basic principle of this cloak. The main limitation of this kind of cloak is that the concealed object cannot receive any information from the outside world, although the outside world cannot detect the concealed object. The advantage of this optical isolation cloak is that the concealed object can be arbitrary (i.e. shape and material) and can move inside the inner boundary of the cloak. In addition to TO, there are many other methods to design an optical isolation cloak, including some geometrical optic methods (e.g. by mirrors). [134] The second kind of cloak designed by TO is a scattering cancellation cloak, which is mainly based on the concept of complementary media and the spatial folding transformation (Figure 4b). [135] The advantage of the scattering cancellation cloak is that the concealed object can still receive electromagnetic waves of the same frequency band from the outside world. However, the scattering cancellation cloak will not work if the concealed object changes (shape or electromagnetic properties). There are also many other ways to design a scattering cancellation cloak besides TO, such as plasmonic shells, [136,137] optimization algorithms, [138,139] or active cloaking. [140] The third type of cloak is a scattering overlapping cloak, which was first proposed by Sun and He in [141] In this kind of cloak, we need a strong scattering object as a background object. The concealed object is set around the background object. Scattering overlapping cloak functions by using the strong scattering of the background object to overlap the weak scattering produced by the object to be concealed (Figure 4c). The main advantages of this scattering overlapping cloak include: (i) The concealed object can still see the outside world (i.e. receive electromagnetic waves). (ii) When the size, shape, or medium of the concealed object changes, the cloak still works. (iii) No need to detect/know the information of the detecting wave in advance. However, this cloak is not an ideal/perfect cloak in theory, as the concealed object may still produce a very weak perturbation in the strong scattering of the background object. All the above three kinds of cloak are full-space cloaks, which is the real sense of invisibility (i.e. the concealed object is in free space). The fourth kind of cloak designed by TO is different from the previous three, it is a half-space carpet cloak. [101,142] The concealed object is under the ground, but not in free space (Figure 4d). The function of a carpet cloak is to make curved ground look like flat ground for an observer above the ground. It is very easy to fabricate such a carpet cloak as its materials parameters are very simple. So far, many experimental demonstrations cloaks are carpet cloaks (i.e. not true invisibility). [ ] Carpet cloaks can also be classified into two subtypes. The first kind of optical carpet cloaks are designed using the OCM method. [101] In the quasi-ocm method, by choosing a suitable transformation, a heuristic approach is made to minimize the induced anisotropy of the parameters. Since the anisotropy is (7 of 27)

8 Figure 5. Different kinds of TO-based optical cloaks. (a) (c) Carpet cloaks designed by the quasiconformal mapping method (Reproduced with permission from Ref. [145]. Copyright (2009) Nature Mater.; Reproduced with permission from Ref. [147]. Copyright (2010) Science; Reproduced with permission from Ref. [149]. Copyright (2011) Nano Lett.). (d) (f) Carpet cloaks designed by homogeneous coordinate transformation (Reproduced with permission from Ref. [143]. Copyright (2011) Phys. Rev. Lett.; Reproduced with permission from Ref. [144]. Copyright (2011) Nature Commun.; Reproduced with permission from Ref. [157]. Copyright (2011) Opt. Express). adequately small, one can ignore it, and only a spatially changing refractive-index distribution is required. The advantage of this cloak is it requires only isotropic parameters, making it easier to fabricate at optical frequencies. [ ] For example, by digging a 2D subwavelength hole lattice with varying density in a silicon slab waveguide, an optical carpet cloak working in the near infrared was demonstrated in 2009 (Figure 5a); [145] In 2010, a threedimensional optical carpet cloak was reported in the near infrared using tailored, dielectric face-centered-cubic woodpile photonic crystals (Figure 5b); [147] A carpet cloak for visible light was also reported in 2011 by etching holes of various sizes in the nitride layer on the deep-subwavelength scale (Figure 5c). [149] The limitation of this kind of the carpet cloak is that there will be a lateral shift in the reflected wave due to the approximations made of the anisotropic parameters. [155] For a practical carpet cloak, the hidden object should be a microscopic object at optical frequencies. However, the hidden regions of this kind of the carpet cloak based on the quasi-ocm are comparable to the wavelength of the operation frequency. The second kind of optical carpet cloaks are designed by homogeneous coordinate transformations. [156] Unlike the quasi-ocm, the homogeneous coordinate transformation separates the virtual space into several triangular regions and applies a linear compression or extension to each region. After that, the constitutive parameters in each region are homogeneous. The cloak needs only anisotropic parameters that simplify the realization of the cloak at optical frequency. In 2011, using a naturally birefringent crystal, two groups, from the Massachusetts Institute of Technology and the University of Birmingham, respectively, proposed a macroscopic visible-light carpet cloak almost simultaneously (Figures 5d and e); [143,144] the same year, a homogeneous carpet cloak at near-infrared frequencies was verified by patterning the silicon layer with nanogratings of an appropriate filling factor (Figure 5f). [157] There is no lateral shift for the reflected wave, and the hidden regions in these cloaks are macroscopic compared to the optical wavelength. However, since it is hard to achieve anisotropic and nonunit permeability at optical frequencies, all of these cloaks only have anisotropic permittivity and can only work for TM-polarized light. It is hard to extend these cloaks into 3D because it is hard to decouple the different polarizations in the 3D case Experimental Realization of an Optical Full-Space Invisibility Cloak Many full-space invisibility cloaks have been experimentally demonstrated in the microwave band (for further reading see reviews on invisibility cloaks [26 28, ] ). In this section, we will focus on typical experimental realizations of full-space invisibility at optical frequencies Polygonal Invisibility Cloak for Visible Light In contrast to the classic optically isolated cloak designed with a spatial extension transformation, where the required parameters are always inhomogeneous and encounter singular values (Eqs. (12) and (13)), a linear polygonal transformation is applied to avoid these limitations. [158] Figure 6 shows how a general m-sided polygonal cloak is designed. An m-sided polygon in Figure 6a is shown as a virtual space with a smaller polygon rotated at an angle of π/m. The space between the two polygons is divided into several triangular segments. These triangular segments in the cloak can be grouped into two types according to their geometries: Segment I and Segment II, with their own local coordinates (u a, v a, w a )and(u b, v b, w b ), respectively (8 of 27)

9 The general m-sided polygonal cloak is composed of many triangular segments with homogeneous and anisotropic materials in physical space, as shown in Figure 6b. On the other hand, since it transforms from a small polygon to a larger one rather than from a point to a circle, it will not require a singular value. In this way, the parameters of the cloak are simplified. Figure 6 shows a 20-sided polygonal cloak, in which the trajectories of the rays are very similar to those in the cylindrical cloak. As far as the trajectory of the ray is concerned, the constitutive parameters of the cloak can be further simplified as a nonmagnetic form: [158] { ε I u = ε/κ2 v,εi v = εκ2 u,μi w = 1, for segment I εu II = ε, εii v = εκ 2,μ II w = 1, for segment II (16) Even with such optimization, there is still difficulty in achieving the required constitutive parameters at optical frequencies. For an experimental demonstration, the cloak is simplified based on the trajectory and the refraction behavior for certain angles. This simplification sacrifices the performance of the cloak, but can still preserve some properties. A simplified hexagonal cloak has been fabricated with naturally birefringent crystals for visible light (Figure 6c). [158] Natural Light Cloak Devices for Large Objects Figure 6. Linear polygonal transformation for a general m-sided polygonal cloak (Reproduced with permission from Ref. [158]. Copyright (2012) Sci. Rep.). (a) Virtual space and (b) Physical space. The red, green and blue lines represent the trajectories of different lights. (c) The top view of the simplified hexagonal cloak. The transformation is applied in all segments along their local coordinate axes with functions: { u a = u a /κ u,v a = κ v v a,w a = w a, for segment I u b = u b /κ, v b = v b,w (14) a = w b, for segment II where κ u = (r 2 r 0 cos(π/m))/(r 2 r 1 cos(π/m)), κ = (r 2 cos π m r 0)/(r 2 cos π m r 1)andκ v = r 1 /r 0 are the compression or extension ratios of the space, respectively. If we consider only the TM case (the magnetic field along the w-direction) and the background medium with permittivity of ε, the parallel-polarized wave-related constitutive parameters of the cloak will be: { ε I u = ε/(κ uκ v ),ε I v = εκ uκ v,μ I w = κ u/κ v, for segment I ε II u = ε/κ, εii v = εκ, μ II w = κ, for segment II. (15) The previously mentioned hexagonal cloak can only work for TM-polarized light because it uses birefringent crystals to realize the anisotropic permittivity. [158] However, it is difficult to realize a TE-polarized or full-polarized cloak using the same method due to the difficulty in achieving anisotropic permeability at optical frequencies. By using the linear polygonal transformation with ray-optics approximations, a cloaking device for large objects in incoherent natural light has been demonstrated (Figure 7a). [32] The transformation of this cloak is similar to that of Eq. (12). As shown in Figures 7b and c, a linear polygonal transformation is applied in different segments, and a polygonal cloak is then designed. To realize such a cloak for incoherent, fully polarized natural light, in the second step, a ray-optics approach is made for parallel light. The anisotropic material segments in the designed cloak in Figure 7c are replaced by isotropic segments in Figure 7d, while the incident rays in the cloak can still bypass the hidden region and turn back to their original path. In this case, unlike previous cloaks where the phase of the light is maintained, the phase-preservation requirement is lifted because the living creatures cannot sense the phase of light. The cloak is fabricated by optical prisms and can work in several directions for the naked eye. [32] Large-Scale Far-Infrared Invisibility Cloak Beside the linear polygonal transformation method, another way to design an optical isolation cloak is the inverse quasiconformal mapping method. Unlike the quasi-ocm method, where the hidden region must be very small because a large hidden region would introduce large anisotropic factors, the inverse quasiconformal mapping method begins with anisotropic virtual space and obtains the physical space with isotropic materials, which can enable a larger hidden region (Figure 8a). [29] As shown in Figures 8b e, the virtual space in Figure 8b is an anisotropic space that is first transformed into two regions with isotropic materials in the intermediate space, as shown in Figure 8c. A rigorous conformal mapping transformation then exists between the (9 of 27)

10 Figure 7. The structure and design method for a large-scale natural light cloaking device (Reproduced with permission from Ref. [32]. Copyright (2013) Nature Commun.). (a) The structure of the experiment setup. (b) Virtual space with an isotropic and homogeneous background. (c) A perfect hexagonal cloak designed from the linear polygonal transformation with homogeneous and anisotropic parameters. (d) A simplified cloak with homogeneous and isotropic parameters. The horizontal rays represent the incident light traveling from left to right, while the red dotted vertical lines represent wavefronts when illumination is coherent. intermediate space and the physical space (Figure 8d). The cloaking material in this physical space will be isotropic. Since the design of this cloak still requires spatially dependent isotropic materials, which are difficult to implement at optical frequencies, one can discretize the inhomogeneous space into several homogeneous spaces. As shown in Figure 8e, the cloak is divided into regions, with homogeneous and isotropic materials. A far-infrared invisibility cloak for large-scale objects has been designed and fabricated using germanium prisms. [29] The cloak works unidirectionally and can hide objects from thermal detection (see Figure 8a). The above-mentioned three optical isolated cloaks for large objects or even living creatures, in some respects, are much closer to the real sense of invisibility. They can hide very large objects at optical and far-infrared frequencies. However, due to the approximations made in the cloak design, these cloaks sacrifice omnidirectional performance and can only work for specific viewing angles. Some impedance mismatch introduced in the realization of the cloak will also cause a slight reflection at the interfaces of the cloak Other Kinds of Invisibility Cloak There are still other optical cloaking methods designed without using TO methods. For instance, a ray-optics cloak was realized in a paraxial limit using a set of optical lenses, [162] a broadband isolated invisibility cloak was achieved in a diffusive light-scattering medium, [163] and an ultrathin carpet invisibility skin cloak for visible light has also been demonstrated with the help of optical metasurfaces. [164] The idea of designing an invisibility cloak for electromagnetic waves with TO has been extended to other physical fields such as acoustic cloaks, [73 75] thermal cloaks, [77 80] cloaks for elastic waves, [ ] matter waves, [168,169] and electronic waves, [170] etc.). DC magnetic cloaks have a clear prospective application, as most metal detectors, such as mine detectors and airport security sense the DC or quasistatic magnetic fields. [102,103] We have experimentally demonstrated a 3D magnetic cloak working for both DC magnetic fields and low-frequency magnetic fields, up to 250 khz, which covers the operation bands of nearly all electromagnetic interference appliances (Figures 9a and b). [171] The practical (10 of 27)

11 potential of such a magnetic cloak was verified by using a commercial metal detector. Recently, Jiang et al. fabricated a 3D room temperature quasistatic magnetic cloak without using superconducting materials, [172] which has potentially real cloaking applications (Figure 9c). 3. New Branches From TO In this section, we will describe two related new branches emerging from classic TO in recent years, namely, optic-null media (ONM) and optical surface transformations (OST). We first introduce the idea of the ONM, where a large volume of real space maps to a single surface in the reference space and then the basic elements of OST that enables the development of meaningful devices with relatively little calculation Optic-Null Medium and Its Applications Background of ONM ONM is an extremely anisotropic, homogeneous medium that can be designed using an extreme stretch in TO. The optical hyperlens designed with TO is an ONM. [44] In a Cartesian coordinate system, we can use the following coordinate transformation to design an ONM: x, x (, 0] x d = x, x [0, d], y = y, z = z. (17) x + d, x [d, ) Figure 8. Principle of inverse procedure of quasiconformal mapping (Reproduced with permission from Ref. [29]. Copyright (2015) Adv. Opt. Mater.). (b) The virtual space with anisotropic parameters. (c) The intermediate space of two regions with isotropic parameters. (d) The physical space of the cloak with homogeneous and spatially dependent parameters. (e) A simplified unidirectional cloak with several homogeneous and inerratic regions. Figures 10a and b show the relation of this coordinate transformation. An extremely thin surface x = 0( 0) in the reference space is stretched to a finite volume in the real space. Since all points on the same line segment (i.e. y = C,0<x <d, C is a constant) in the real space correspond to one point (i.e. x = 0, y = C) in the reference space, all of these points are equivalent points. If we set one line current at the input surface of such an ONM, we can obtain its image at the output surface. Figure 10c shows the Figure 9. Quasistatic magnetic cloak to hide metals (reprinted from Refs. [171] and [172] with permission). (a) The schematic of the cloak with superconductor materials (Reproduced with permission from Ref. [171]. Copyright (2015) Nature Commun.). It consists of an inner superconductor shell (R 1 r < R 2 ), colored in black, and an outer ferromagnetic shell (R 2 r R 3 ), colored in brown. (b) The measured relative change of the z-component magnetic field at the operating frequency varies from 5 Hz to 250 khz. The inset plots the real permeability spectrum of the FM material measured at room temperature (Reproduced with permission from Ref. [171]. Copyright (2015) Nature Commun.). (c) The schematic of the measurement setup for the 3D room-temperature quasistatic magnetic cloak using a commercial metal scanner (Reproduced with permission from Ref. [172. Copyright (2017) NPG Asia Mater.). A temperature meter is placed in the lower right corner (11 of 27)

12 Figure 10. (a) and (b) show the transformation relation (i.e. Eq. (17)) between the reference space (a) and the real space (b) to design an ONM in a Cartesian coordinate system. A very thin volume (approximate surface with the normal vector along the x-direction when its thickness 0) in the reference space is extremely extended/stretched to a finite volume with thickness d along the x -direction in the real space. The volume in the real space is filled by the ONM described by Eq. (18) (i.e. the main axis of the ONM is in accordance with the direction of the stretching). (c) 2D numerical simulation results. We plot the snapshot of the electric field s z -component (TE-wave case). If we set a line current source with unit amplitude at one surface of the ONM in Eq. (18), we can obtain its image on the other side of the ONM. (d) and (e) show the transformation relation (Eq. (19)) between the reference space (d) and the real space (e) to design an ONM in a cylindrical coordinate system. (f) 2D numerical simulation results when we set two coherent line currents with unit amplitude at the inner surface (i.e. r = R 1 = λ 0 ) of the ONM in Eq. (20) with a subwavelength separation (i.e. λ 0 /5). We plot the normalized amplitude of the electric field s z -component. At the outer surface of the ONM (the optical hyperlens) r = R 2 = 5λ 0, there are two images with an amplified separation. numerical simulation of the ONM s imaging ability. The relative permittivity and permeability of such an ONM can be calculated by TO (Eqs. (1) and (2)): ( d diag ε = μ =, d, ) 0 diag(, 0, 0), x [0, d] d. 1, x (, 0) (d, ) (18) We should note that ONM can achieve superresolution imaging that can break the diffraction limit, as such a highly anisotropic medium can convert evanescent waves to propagating waves on the input surface of the ONM and back to evanescent waves at the output surface. [173,174] We often use an ONM in a cylindrical coordinate system for imaging applications, since it provides a predetermined magnification. Figures 10d and e show how to design an ONM in a cylindrical coordinate system with the following coordinate transformation: r, r (R 2, ) R 2 R 1 r = r + R 2 R 1 R 2, r [R 1, R 2 ], θ = θ,z = z. R 1 R 2 r, r (0, R 1 ) (19) Combining with Eqs. (2) and (4), we can obtain the medium of such an ONM in a cylindrical coordinate system: ε cy = μ cy ( ( ) ) ( R2 2 0 diag 1, 1, diag 1, 1, R 1 ( = diag P, 1 ( ) ) 2 P, 0 P diag(, 0, 0), d ( R2 r [R 1, R 2 ] 1, r [R 2, ) R 1 ) ) 2, r [0, R 1 ], (20) where P = 1+((R 2 R 1 )/ 1)R 2 /r. The subscript cy indicates that the quantity is expressed in a cylindrical coordinate system. Note that the medium in the region R 1 <r <R 2 is the ONM in the cylindrical coordinate system. The superresolution imaging ability of such an ONM is shown in Figure 10f. Note that the magnification factor of this lens is R 2 /R 1 (the geometrical ratio of its outer radius to its inner radius). In general, an ONM can be in an arbitrary coordinate system. There is a main axis of an ONM, whose relative permittivity and permeability are extremely large along the main axis and nearly zero in all orthogonal directions. ONMs have been experimentally demonstrated using different metamaterials. [175,176] (12 of 27)

13 Figure 11. 2D numerical simulation results for the optical overlapping illusion using ONM (Reproduced with permission from Ref. [180]. Copyright (2016) Sci. Rep.). We plot the absolute value of the electric field s z -component (TE-polarization case). (a) (c): we set one, two and three inphase coherent line currents (indicated by black arrows) with unit amplitude at the equivalent positions (i.e. in the same line y = Constant) inside a rectangular ONM described by Eq. (18). The size of the rectangular ONM is chosen by a length d = 8λ 0 /3 along the x -direction and height h = 2λ 0 /3 along the y -direction. Note that the electric-field distributions are exactly the same, but color bars are amplified by factors of 1, 2, and 3 from (a) to (c), which verifies the overlapping effect. (d) (f): we simply remove the ONM while keeping the line currents (indicated by black arrows) corresponding to (a) (c). Since the ONMs are removed, the electric fields are totally different from (d) to (f). A metallic plate with fractal holes has been theoretically and experimentally demonstrated to be an ONM, with hyperlensing in a cylindrical coordinate system. [175] A metallic slit array structure serves as an ONM for electromagnetic wave concentration at microwave frequencies when the Fabry Pérot (FP) resonance condition is satisfied. [176] These experimental results also show that the loss in real metals does not influence the performance of the ONM. The idea of creating ONM by an extreme stretching coordinate transformation has been extended to other physical fields, it is known by other names such as magnetic hoses for DC magnetic fields [109,110] and heat hoses for the thermal fields. [177] From the perspective of TO, they are essentially the same idea. For a DC magnetic field, we can use the same coordinate transformation Eqs. (17) and (19) to design a DC magnetic hose in Cartesian and cylindrical coordinate systems, respectively. We only need the relative permeability in Eqs. (18) and (20) to realize such a DC magnetic hose. DC metamaterials (combination of a superconductor with a permeability of nearly zero and a ferromagnetic material with extremely large permeability [178,179] ) have been utilized to realize such a DC magnetic hose for various applications, including long-distance DC magnetic field transfers, [109] DC magnetic field concentrators, [105,106] and DC magnetic loops. [110] Analogously, a thermal hose whose thermal conductivity is expressed by Eq. (18) can transfer thermal energy over arbitrarily long distances. [177] ONMs for Optical Overlapping Illusions An optical overlapping illusion is possible since many different spatial points in the real space can correspond to one single point in the reference space. In other words, all of these points in the real space are equivalent points. We can simply set N optical sources at these equivalent points and achieve a radiation pattern exactly similar to if we had set one single optical source in the reference space with an N-fold amplitude). There are an infinite number of equivalent points in an ONM. For the ONM in a Cartesian coordinate system (Eq. (18)), any point on the same line segment (y = C, 0<x <d, C is a constant) in the real space corresponds to one point (x = 0, y = C) in the reference space. We can use this feature to achieve an optical overlapping illusion (see Figure 11). [180] ONM (Figure 11) has several advantages when compared with other methods to achieve an optical overlapping illusion. [ ] We can achieve an optical overlapping illusion with arbitrary power magnification with a desired radiation pattern using the ONM. Changing the boundary of an ONM can modulate the radiation pattern of the overlapped sources. The overlapping illusion produced by the ONM is not sensitive to loss. We have unlimited equivalent points in an ONM so the number of sources can be changed without redesigning the device. Besides the power combination, ONMs have also been applied to extend the scanning angle of a phased array antenna (13 of 27)

14 in a predesigned manner (i.e. the relation between the input angle and the output angle of the radome can be analytically designed). [186,187] A similar idea can also be extended to other physical fields. For a DC magnetic field, a DC magnetic hose is similar to ONM. [109] If we make a compression finite embedded transformation (FET) inside a DC magnetic hose, we can obtain a DC magnetic funnel that can concentrate the input DC magnetic flux. [188] 3.2. The Optical Surface Transformation and Its Applications Originating from the idea of equivalent surfaces in ONM, a general theory (OST) that can build an equivalence relation between two surfaces of arbitrary shapes was developed. [189,190] In this section, we will first briefly review the basic theory of OST and then show many novel optical devices designed by OST Basic Theory of OST OST is another new branch of TO. [189,190] All of the above devices designed by TO require a proper coordinate transformation to be found/designed (i.e. an analytical calculation is required). Transformation media designed by TO are often complicated requiring inhomogeneous, anisotropic, or magnetic materials. The purpose of proposing an OST, whose basic principle is still TO theory, is to design novel optical devices in a much simpler manner. In this section, we introduce how to obtain the OST from TO and then explain the idea of equivalent surfaces. In later sections, we will show how to use OST to design various kinds of novel optical devices. In previous sections, we have introduced the idea of equivalent points, where multiple points in the real space corresponding to the same point in the reference space. Now we introduce the idea of equivalent surfaces, where multiple surfaces in the real space (often linked by ONMs) correspond to the same surface in the reference space. The basic element to make an OST is the ONM that we have also introduced in Section 3.1. Two arbitrarily shaped surfaces S 1 and S 2 can be linked by two ONMs with the main axes along the x -andy -directions, respectively, in a 2D plane (Figure 12a). We first project S 1 along the x -direction to a plane S 1 and project S 2 along the y -direction to a plane S 2.Next, if we continue projecting S 1 along the x -direction and S 2 along the y -direction, a common plane S is obtained, which is the diagonal of the rectangle outlined by the shifted S 1 and S 2. We fill the ONMs between the above surfaces whose main axes directions consist of the projecting directions (ONMs colored by green and yellow correspond to the main axes along x and y, respectively, in Figure 12a. We use the label X or Y to mean that the main axis of this ONM is along the x -ory -direction throughout the paper. Next, we will show that S 1, S 1,andS, linked by an ONM with the main axis along the x -direction, are equivalent surfaces (any point on one surface can have a corresponding point on its equivalent surface, which corresponds to one single point in the reference space). For simplicity, we only show that S 1 and S 1 are equivalent surfaces, as S can be treated as a special case when the arbitrary shaped surface S 1 is chosen as another plane. As shown in Figure 12b, S 1 is divided into many small plane elements S i. Projecting S i along the x -direction onto the plane S 1,weobtain its corresponding small plane element S i. Next, we use TO to prove that two small plane elements S i linked by an ONM correspond to one single small plane element in the reference space. As shown in Figure 12c, two small plane elements S i and S i in the real space correspond to the same small plane element S i in the reference space when θ 0 90 [deg] and 0. The coordinate transformation is given by: d x, x [0, d] x = tan θ 0 (x ) + d, x [d, d + S i ] ; y = y; z = z. tan θ tan θ x else (21) The transformation medium in the trapezoid region of the real space can be obtained by combing Eqs. (1), (2) and (21) with θ 0 90 [deg] and 0: diag(, 0, 0), x [0, d + S i ] ε = μ = tan θ. (22) 1, else Equation (22) shows that the transformation medium in the trapezoid region of the real space is the ONM with the main axis along the x -direction. We can divide the green region between S 1 and S 1 in Figure 12a into many such small trapezoid regions (Figure 12b), in which the ONMs with the main axes along the x -direction are filled. Each small trapezoid region corresponds to a single surface in the reference space, and hence surfaces S 1 and S 1, linked by the green ONM in the real space, correspond to a common single surface in the reference space. The transformation medium linking S 1 and S in Figure 12a is the same as the medium that links S 1 and S 1, as surface S can be treated as the special case where surface S 1 is chosen as a plane. Thus, we have shown that S 1, S 1,andS linking by the ONM with the main axis along the x -direction are equivalent surfaces by TO. We can use a similar method to prove that S 2, S 2,andS, linkedbyan ONM with the main axis along the y -direction (colored by yellow in Figure 12a) are also equivalent surfaces. Since the surface S is the common surface between these two sets of equivalent surfaces, we can conclude that surfaces S 1 and S 2 are equivalent surfaces. We can always make two arbitrary shaped surfaces S 1 and S 2 as an equivalent surface by projecting them to a common plane (S in Figure 12a) and filling ONMs with the main axes directions the same as the projecting directions between these surfaces. The physical meaning of the equivalent surface is that any point P 1 on S 1 can find its corresponding point P 2 on S 2, and two points, P 1 and P 2, correspond to the same point P 0 in the reference space (Figure 12d). In other words, any optical information such as the electric-field distribution, the optical wavefront distribution, optical sources distributions, etc. on surface S 1 can be exactly transformed to surface S 2. This is the key point of an OST: we can simply design the geometrical shapes of the input and output (14 of 27)

15 Figure 12. (Reproduced with permission from Ref. [190]. Copyright (2015) Sci. Rep.) (a) A general outline for linking two arbitrarily shaped surfaces S 1 and S 2 by ONMs (i.e. making them equivalent surfaces). The yellow and green regions are filled by ONMs with main axes along the x - and y -directions (labeled X and Y), respectively. The function of the ONM is to project the surface along its main axis direction, and hence S 1, S 1, and S are equivalent surfaces. Similarly, S 2, S 2, and S are equivalent surfaces. (b) we divide the region between S 1 and S 1 into many small subregions along x (the main axis direction of the ONM filled between S 1 and S 1 ). The surfaces S 1 and S 1 are divided into many subsurface element pairs ( S i and S i ). (c) Multiple small surface element pairs in the real space can correspond to the same small surface S i in the reference space by Eq. (21) when θ 0 90 o and 0. (d) 2D numerical simulation results (TE-wave case): we plot the absolute value of the normalized electric field s z -component. The shapes of S 1 and S 2 are chosen arbitrarily. If we set a line current with unit amplitude on surface S 1 (purple line), we will get its image at S 2 (brown line). surfaces of the devices without using any analytical coordinate transformations and tensor calculations to achieve various optical functions (we will give many examples later). All we need is one homogeneous, anisotropic medium, namely ONM, to realize all devices designed by OST. The function of an ONM can be treated as a projecting transformation along the direction consistent with the direction of its main axis Some Typical Devices Designed by OST Any optical information on one surface (S 1 in Figure 12a) can be transferred to its equivalent surface (S 2 in Figure 12a) by choosing the appropriate optical projections. Note that such optical projections are achieved by filling the ONMs in the suitable regions in Figure 12a. We can control the optical wavefront distribution by controlling the shape of the two surfaces. The geometrical shape of the device s input surface is chosen as the shape of the incident wave s wavefront. The geometrical shape of the device s output surface is designed as the shape of the wavefronts that we want to achieve. Once the geometrical shapes of the input surface S 1 and output surface S 2 of our device are fixed, we can use the above OST to design the medium between S 1 and S 2 (i.e. how to fill the ONMs, as shown in Figure 12a). Figure 13 shows many different examples of wavefront converters. With the help of OST, we can also modulate the amplitude of the incident beam by choosing a proper geometrical shape of an interface function f(x) (Figure 14). [191] The OCM or quasi-ocm has been applied to change the circular focusing surface of the Luneburg lens to a plane. [ ] However, numerical calculations (generation of conformal or quasiconformal grids) or analytical coordinate transformations are required. Once the geometrical size of the device changes, we must make new calculations. We can simply convert the shape of the focusing surface of the Luneburg lens, Maxwell s fish-eye lens, and hyperlens by filling an ONM between the original focusing surface and a plane (the reshaped new focusing surface). [190] Novel open optical cavities that have small mode volumes (subwavelength confinement) and high Q-factors at the same time are realized by TO. [ ] Based on the OST, we can also design an open optical resonator of an arbitrary shape with a small mode volume and high Q-factor in a much simpler way. [199] (15 of 27)

16 Figure 13. 2D numerical simulation results for the TE-wave case. (a) The input and output surfaces are chosen as a cylindrical surface and plane, respectively (Reproduced with permission from Ref. [190]. Copyright (2015) Sci. Rep.). The region between the input and output surfaces is filled by ONM (the main axis is along the x-direction). We set a unit line current source in the center of the cylindrical surface and obtain a plane wave at the output surface. The black arrow indicates the propagation direction of the output plane wave. (b) The input and output surfaces are both cylindrical surfaces (Reproduced with permission from Ref. [190]. Copyright (2015) Sci. Rep.). In this case, we can convert a diverging cylindrical wave produced by the line current source (the left red point) to a converging cylindrical wave (converged at the right red point). (c) The Bessel beam converter (Reproduced with permission from Ref. [191]. Copyright (2016) J. Opt. Soc. Am. B): the ONM (amplified on the right inserted subfigure) with a conical output surface of a tilt angle θ to create a converged conical wavefront (i.e., a zeroth-order Bessel beam) The Scattering Cover-up Cloak by an OST As we have introduced in Section 2.1, there are three kinds of fullspace invisibility cloak (optical isolation cloak, scattering cancellation cloak, and scattering cover-up cloak). For a scattering coverup cloak, we need a background object (Figure 4c) with strong scattering. Such a cloak can be simply designed with the help of an OST. We have a cylindrical PEC as the real background object (Figure 15a) in the center of an ONM whose main axis is along the radial direction (the medium in the region R 1 <r <R 2 in Eq. (20)). The concealed PEC objects of arbitrary shapes are also set within the ONM (the moon and heart in Figure 15a). According to OST, all boundaries connected by the ONM here are equivalent surfaces. Whatever the shapes and locations of the concealed objects inside the ONM, the system (the concealed objects, the background PEC in the center, and the ONM) appears like a large PEC with the same boundary as the outside boundary of the ONM (Figure 15b). This means that when a detecting plane wave is incident onto the cloak (Figure 15a) and a large PEC background object (PEC in reference space in Figure 15b), the scattering electric fields should be exactly the same. [200] An outside observer can only see the large PEC background object without seeing the concealed objects (the moon and heart in Figure 15a) inside the cloak. We should note that we have two background objects: one is the real background object wrapped inside the cloak (the PEC with a small radius R 1 in the center of the ONM in Figure 15a). The other is the virtual background the object with a larger size in the reference space (the PEC with a larger radius R 2 in free space in Figure 15b). In practice, we need the real background object to produce the cloaking effect together with the ONM. What observers can see/detect is the virtual background object. Numerical simulations in Figures 16a and b verify the performance of the scattering cover-up cloak. If we apply the cloak (the ONM) around the concealed PEC objects (i.e. one is the circular column on the left and the other is the rectangular column on the top in Figure 16f and the real background PEC object, the absolute values of the electric-field difference, as compared to the case where only the virtual background PEC object exists, are all nearly zero in the whole space (Figure 16a). However, if we remove the cloak while keeping the real PEC background objects and the concealed PEC objects unchanged, the total electric field is greatly different (Figure 16b). It is also worth mentioning that, even if we replace the PEC concealed objects in Figure 16a by dielectric objects with the same geometrical shapes, the cloak still keeps a certain effect (Figure 16c with cloak and Figure 16d without cloak). In this case, some electric field can enter the concealed regions (the concealed objects can still receive information from the outside world without being seen). Compared to other methods to achieve invisibility, this scattering cover-up cloak designed by an OST has many interesting features: i) The concealed objects can see the outside world without being detected (electromagnetic waves can enter into the (16 of 27)

17 Figure 14. The basic idea of modulating the amplitude of optical beams by an OST (Reproduced with permission from Ref. [191]. Copyright (2016) J. Opt. Soc. Am. B). (a) The basic structure composed of ONMs to achieve amplitude modulation. The regions colored red and green are ONMs with main axes along the x- andy-directions, respectively. (b) (e) are 2D numerical simulation results for the TE-polarization case. (b) and (c): We plot the snapshot of the electric field s z-component distribution when a Gaussian beam is incident from the bottom onto the device to obtain a hollow beam and an asymmetric beam, respectively, at the output surface of the device. (c) and (e) are the amplitudes of the electric field s z-component at the output surface of the device in (b) and (d), respectively. concealed regions). ii) The concealed objects can be arbitrarily shaped and can also move freely within the cloak (if the shapes of the concealed objects change, the cloak still works). iii) Only one homogenous anisotropic medium (i.e. the ONM described by Eq. (20) in the region R 1 <r <R 2 ) is required to achieve such a cloak. iv) There is no need to know any information of the detecting wave in advance. v) If the concealed objects are PEC, the cloaking effect is perfect in theory. If the concealed objects are dielectric materials, such a scattering cover-up cloak by the ONM can still work, although with certain performance degradation. vi) Such a cloak is a full-space invisibility illusion, which cannot work without a PEC background object Realizing the Devices Designed by an OST Without ONMs As we have introduced, the ONM (a highly anisotropic medium) is the necessary element to realize the devices designed by an OST. However, there are only some metamaterials working at microwave frequencies that can realize the ONM. [175,176] It is still difficult to realize the ONM in other frequency bands such as the optical band. We have introduced the idea of the anti-onm to effectively realize the devices designed by an OST with some lowanisotropy media). [201] The basic idea is shown in Figures 17aand b. In the reference space in Figure 17a, we have an ONM filled in the region B 2 B 3 D 3 D 2 D 3 B 3 with the yellow boundary, and all (17 of 27)

18 4.1. Other Classic Devices Designed by TO Figure 15. (Reproduced with permission from Ref. [200]. Copyright (2016) Sci. Rep.) (a) The whole structure in the real space, including a real background object (e.g. the small blue circle PEC with a radius R 1 ), the concealed objects (e.g. the green moon and red heart), and the cloak (the ONM, colored yellow). To an outside observer, the scattering field produced by the system in (a) is exactly the same as the scattering field produced by a large PEC (the virtual background object with a radius R 2 )in (b). other regions are free space. In the real space in Figure 17b, we only need four regions filled by homogeneous low-anisotropy media and all other regions filled with free space to realize the same optical effect produced by an ONM in the reference space for the region outside the rectangle A 2 B 2 C 2 D 2. For example, we can set three line current sources inside the ONM in the reference space to achieve a power combination effect in Figure 17c. Equivalently, we can also set three line current sources in the free space inside the low anisotropic shell in Figure 17b to achieve the same electric-field distribution (Figure 17d). The low anisotropy medium in each region, with different colors in Figure 17b, is given by: [201] ( M 2 + N 2) N M M 0 ε = μ = N 1 M M M, (23) where { M = P, N Q = sgn(x )sgn(y ) s Q Q = s + P, P = a + d/2 b, s = (D d) /2. (24) where sgn is a signum function (i.e. sgn(x ) = 1forx >0, and sgn(x ) = 1 for x <0) to distinguish different regions. The yellow boundary of the ONM can also be any other geometrical shape designed by an OST (more examples are given in Ref. [201]). 4. Other Devices and Applications by TO We have reviewed invisibility cloaks designed by TO and the theory of OST derived from TO in detail. In this section, we will briefly show many other devices designed by TO and other applications of TO. Among other devices designed by TO, we will focus on one typical device, an electromagnetic-wave rotator, and show a few novel applications of such a classic device. There are two main kinds of devices designed by TO. One is designed by a continuous coordinate transformation in the whole space. In this case, these devices do not influence the EM wave outside the device and only change the path of the light inside the device (e.g. invisibility cloaks, [20] wave concentrators, [22] and rotators [35,36] ). A wave concentrator can concentrate the incoming electromagnetic wave to a given region by a spatial compression transformation, [22] which may have potential applications in energy-harvesting, solar cells, and wireless power transfer systems. Negative refractive index materials (NRIMs) [ ] can also be designed by TO using a spatial folding transformation. [19] By applying a spatial folding transformation, different points in the real space can correspond to one single point in the reference space. In other words, perfect imaging can be achieved by such NRIMs. Such a perfect imaging device designed by TO with NRIMs is often referred to as a superlens. [202] Many other illusion optical devices are often related with such spatial folding transformations and NRIMs, such as invisibility gates, [52] remote waveguide connectors, [62] superscatterers and superabsorbers, [49 51] optical shifters, [ ] etc. The second main type of TO devices are designed by the finite embedded transformation (FET), where the coordinate transformation is applied within a finite domain and not necessarily continuous at the boundary of this domain. [39] These devices can control the wavefront or polarization of the incident beam. Beam shifters can produce a predesigned spatial displacement on the input beam with an arbitrary incident angle. [39] Beam splitters can split the incident beam into two new beams [40] and can also be utilized as a beam combiner, as the light path is reversible. [207] Beam compressors and expanders can compress and expand the light spot size of the incident beams, respectively. [41] Flat focusing lenses can also be designed by FET. [208] In a FET, the metrics of space are not necessarily recovered into those of the original reference space, which means there may be some reflections of these devices. [ ] By choosing different kinds of FETs, researchers have achieved various polarization-controlling devices, including polarization splitters, [40] polarization rotators, [212] and polarization controllers. [213] All these devices can also be applied to integrated photonic systems as various elements such as collimators, benders, splitters, and adapters on silicon waveguides). [214]. Many devices designed by TO have been extended to other physical fields for different applications (see some recent reviews for details, Refs. [69,71] and [72]). Multiphysical devices that control two different physical fields at the same time in a predesigned manner can also be designed using a coordinate transformation method [215] and have been experimentally demonstrated. [216] 4.2. An Electromagnetic Wave Rotator A wave rotator is another typical classic device designed by TO that has been applied to beam-angle steering [35,36] and even cloaking effects. [37] We can use the following spatial rotation transformation to achieve an optical rotator: (18 of 27)

19 Figure 16. (Reproduced with permission from Ref. [200]. Copyright (2016) Sci. Rep.) (a) (e) are 2D numerical simulation results for TE-wave case. (f) shows the shapes and locations of the concealed objects (e.g. the red rectangle and green cylinder) in the above simulations. (a) and (b): the concealed objects are PEC. (a) We plot the absolute value of the electric-field difference between the case where the whole system in (f) exists and the case where only the virtual background object exists. (b) We simply remove the cloak and keep other things the same as (a). (c) and (d) The concealed objects are dielectric media (e.g. the left cylinder in (f) with ε r = 1andμ r = 40 and the upper rectangular cylinder in (f) with ε r = 10 and μ r = 1) with and without the cloak, respectively. (e) We magnify the total electric field in (c) to verify that the detecting field can enter into the concealed region. r = r θ,r b θ = θ + b r b a α, a < r < b, (25) θ + α, r a z = z where a and b are radii of the inner and outer boundary of the rotator. α is the rotation angle from the outer boundary r = b to the inner boundary r = a. The relative permittivity and permeability of the rotator in the cylindrical coordinate system can be calculated by TO (i.e. Eqs. (2) and (4)): 1 B 0 ε r r ε r θ 0 ε cy = μ cy = B 1 + B 2 0 := ε r θ ε θ θ 0, a < r < b, , else (26) (19 of 27)

20 Figure 17. (Reproduced with permission from Ref. [201]. Copyright (2016) Phys. Rev. B) (a) and (b): The transformation relations between the reference space (a) and the real space (b). All white regions are free space in both (a) and (b). The region in the yellow boundary in the reference space is filled by the ONM with its main axis along the x-direction. Four different regions, labeled by different colors, in the real space correspond to the low anisotropic media given by Eq. (23). (c) and (d) are 2D numerical simulation results for TE-polarization case. (c) We set three line current sources within the ONM in (a) whose locations are indicated by the black arrows. (d) we set three line currents in free space inside the low anisotropic shell in (b). The electricfield distributions outside the shell region (i.e. the big black square) in (c) and (d) are exactly the same, which means that the function of such a low anisotropic shell is equivalent to the ONM. where B = r α/(a b). If we choose α = π [rad], this rotator can give a beam effectively a rotation of 180 degrees (see Figure 18a). Our recent study found that such a rotator with α = π can reverse the direction of space (see Figures 18b e) and achieve an inverse Doppler effect without any NRIMs [38]. Figures 18b e show the electric field produced by a line current source at (x 0,0)inside the rotator with α = π is exactly the same as the field produced by a line current source at ( x 0, 0) in free space without the rotator. This means that the direction of x is reversed inside and outside the rotator, which implies that an inverse Doppler effect can be achieved (for more details, see Ref. [38]) Other Applications of TO In addition to designing novel optical devices, TO has many other kinds of applications. We can summarize its other applications into following aspects: Simplifying the material parameters of other optical devices is another application of TO. The most important example is removing the singularity of optical devices with TO. [ ] An Eaton lens that is a full-angle retroreflector has been proposed for many years. [12] However, it is challenging to fabricate such a lens due to the singular point in its center (the reflective index is infinitely large here). In 2010, an Eaton lens with the singularity removed using TO was experimentally demonstrated for the first time. [11] TO has also been utilized to analytically study the absorption spectrum, spatial distribution of modes, and van der Waals forces of plasmonic particles with different geometrical shapes including sharp corners. [ ] Many broadband plasmonic singular nanostructures have been proposed with TO under a quasilimit approximation, such as crescent-shaped nanoparticles, [223,224] nanowires, [ ] sharp edges, [228] and nanospheres. [229] TO can reshape the geometrical shape of optical devices. For examples, TO has been utilized to reshape the focusing plane of a Luneburg lens. [ ] The focusing plane of a Luneburg lens is spherical in 3D or circular in 2D, which makes it hard to integrate with planar CCD arrays. 2D and 3D Luneburg lenses with flat focusing planes have been designed and experimentally demonstrated using TO and metamaterials. [ ] Thesameideahas also been applied to other kinds of lenses, such as Maxwell s fisheye lenses. [230] In addition to the above aspects, TO has many other applications, such as calculating the optical force, [231] designing gratings, [ ] designing spontaneous emissions, [235] reverse ray tracings, [236] designing perfect matched layers in numerical simulations, [237] etc. In recent years, more and more new (20 of 27)

21 Figure 18. 2D numerical simulation results for electromagnetic-wave rotator (Reproduced with permission from Ref. [38]. Copyright (2016) Eur. J. Phys.). We plot the snapshot of the electric field s z -component (TE-wave polarization). (a) A Gaussian beam incident onto the 180-degree rotator from the left. The black arrow indicates the propagation direction of the wavefront. Note that the propagation direction is reversed inside the rotator. (b) and (c): We set a line current with unit amplitude inside the 180-degree rotator. The spatial positions of line currents are at +0.5λ 0 and +2λ 0 from the center on the x -axis, respectively. λ 0 is the free-space wavelength. (d) and (e): We set a line current with unit amplitude in free space without the rotator. The spatial positions of line currents are at 0.5λ 0 and 2λ 0 from the center of the x -axis, respectively. To make a better comparison, we keep the black circles, which indicate the corresponding location of the rotator in (b) and (c). Note that the total electric-field distributions are exactly the same outside the larger black circle in (a) and (d) (also in (c) and (e)). branches are emerging from TO, such as TO with graphene, [238] TO with photonic crystals, [239] TO with meta-surfaces, [240] TO with Fourier optics, [241,242] TO for lossy media by complex coordinate transformations, [ ] etc. Such a coordinate transformation method that can link physical quantities in two spaces will lead the trends in future optical design and multiphysical fields design. [215,216] Surface plasmon polaritons (SPPs), in which photonics are coupled to free electrons in metals, has been another hot top in recent years. [ ] Plasmonic devices can confine the light on the nanoscale, which is leading the trend in nano-optical technologies, such as superresolution imaging, nanofocusing, nanooptical circuitry, and lithography. TO has also been applied to control the propagation of SPPs and design novel plasmonic devices (see Figure 19). [ ] SPPs are surface modes between metal and dielectric medium, whose energy most resides in the dielectric medium. That is why there is only the need to transform the dielectric medium by TO and keep metal unchanged in designing plasmonic devices. [249,250] 5. Conclusions and Outlook TO is leading the way to design novel optical/electromagnetic devices that cannot be designed by traditional methods. Similar ideas have been extended to other physical fields and waves, such as acoustic waves, thermal fields, DC magnetic/electric fields, elastic waves, quantum waves, etc. As the development of nanophotonics and new materials continues, many novel devices with fascinating predesigned functions that could never have been imagined before will become reality. In this review, we began with the background and basic theoretical formulas of TO and then introduced the most exciting application of TO, the invisibility cloak. Later, we focused on two new important branches from TO (ONM and OST), which would lead to a much simpler and easier way to design new optical and photonic devices in the future. Finally, we systematically reviewed many other devices and applications/extensions of TO. As TO theory matures, more and more branches of TO are expected to emerge in the future. Some branches of TO focus (21 of 27)

22 We are grateful to Dr. Julian Evans for valuable discussions. This work is partially supported by the National Natural Science Foundation of China (Nos , , , , , and ), the National Natural Science Foundation of China for Young Scholars (No ), the National High Technology Research and Development Program (863 Program) of China (No. 2012AA030402), the Program of Zhejiang Leading Team of Science and Technology Innovation, the Top-Notch Young Talents Program of China, the Postdoctoral Science Foundation of China (No. 2013M541774, 2015M581930, and 2017T100430), the fundamental research funds for the central universities (No. 2017FZA5001), the Preferred Postdoctoral Research Project Funded by Zhejiang Province (No. BSH ), Swedish VR grant (# ) and AOARD. Conflict of Interest The authors have declared no conflict of interest. Keywords Transformation optics, Invisibility cloak, Optical surface transformation, Optic-null medium, Metamaterials Figure 19. Plasmonic devices designed by TO: (a) cloak for SPPs (Reproduced with permission from Ref. [249]. Copyright (2010) Nano Lett). (b) Carpet cloak for SPPs (Reproduced with permission from Ref. [252]. Copyright (2010) Opt. Express). (c) 180 o waveguide bender for SPPs (Reproduced with permission from Ref. [250]. Copyright (2010) Nano Lett). (d) Luneburg lens for SPPs (Reproduced with permission from Ref. [251]. Copyright (2011) Nature Nanotechnol.). on making the inverse design process simpler, such as OST and ONM mentioned in this review, which will be further improved in the future. Some branches of TO will enable us to design more kinds of media with the coordinate transformation method, such as chiral media, time-varying media, nonimpedance-matched media, bianisotropic media. For devices designed by TO, there are still many promising, but unexplored avenues to study. With the development of new artificial materials and technologies, more and more devices designed by TO can be fabricated or improved upon. An interesting subject is how to realize the same function of TO-based devices by some other means. For example, devices based on the space-folding transformation, such as superscatterers, [49 52] antimagnets, [112] and moving-object projectors, [203] currently require a negative refractive index. Can we achieve the same function of these devices without the requirement for any negative permittivity or permeability by some other methods? More studies are needed on this topic. Besides the classic devices (invisibility cloaks, concentrators, rotators) designed by TO, more and more novel devices with fascinating functions (not limited to optics) will be designed or even experimentally demonstrated in the future, such as time-varying devices, multifunctional tunable devices, active devices, etc. Acknowledgments Received: February 6, 2017 Revised: April 29, 2017 Published online: September 29, 2017 [1] J. D. Jackson, Classical Electrodynamics (Wiley, 1999). [2] M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (CUP Archive, 2000). [3] W. C. Chew, Waves and Fields in Inhomogeneous Media (Vol. 522). (IEEE Press, 1995). [4] Y. A. Kravtsov and Y. I. Orlov, Geometrical Optics of Inhomogeneous Media. (Springer-Verlag, 1990). [5] J. C. Maxwell, Camb. Dublin Math. J. 8, 188 (1854). [6] C. T. Tai, Maxwell fish-eye treated by Maxwell equations, Nature 182, 1600 (1958). [7] A. S. Gutman, Modified luneberg lens. J. Appl. Phys. 25, (1954). [8] R.K.Luneburg,Mathematical Theory of Optics (University of California, 1964). [9] S. P. Morgan, General solution of the Luneburg lens problem, J. Appl. Phys. 29, (1958). [10] Y. L. Loo, Y. Yang, N. Wang, Y. G. Ma, and C. K. Ong, Broadband microwave Luneburg lens made of gradient index metamaterials. J. Opt.Soc.Am.A29, (2012). [11] Y. G. Ma, C. K. Ong, T. Tyc, and U. Leonhardt, An omnidirectional retroreflector based on the transmutation of dielectric singularities. Nature Mater. 8, (2009). [12] J. E. Eaton, Trans. IRE Antennas Propag. 4, 66 (1952). [13] M. Yin, X. Y. Tian, and L. L. Wu, All-dielectric three-dimensional broadband Eaton lens with large refractive index range. Appl. Phys. Lett. 104, (2014). [14] A. L. Mikaelian, Application of stratified medium for waves focusing. Dokl. Akad. Nauk SSSR 81, (1951). [15] A. L. Mikaelian, General method of inhomogeneous media calculation by the given ray traces. Dokl. Akad. 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