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Received: Nov. 11, 2018

Accepted: Dec. 6, 2018

Posted: Mar. 14, 2019

Published Online: Feb. 18, 2019

The Author Email: Pendry John B. (j.pendry@imperial.ac.uk), Luo Yu (luoyu@ntu.edu.sg)

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Jingjing Zhang, John B. Pendry, Yu Luo. Transformation optics from macroscopic to nanoscale regimes: a review[J]. Advanced Photonics, 2019, 1(1):014001

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###### Advanced Photonics, Vol. 1, Issue 1, 014001 (2019)

## Transformation optics from macroscopic to nanoscale regimes: a review

Jingjing Zhang^{1}, John B. Pendry^{2,*}, and Yu Luo^{1,*}

Author Affiliations

^{1}Nanyang Technological University, School of Electrical and Electronic Engineering, Singapore^{2}Imperial College London, Blackett Laboratory, Department of Physics, London, United Kingdom

## Abstract

Transformation optics is a mathematical method that is based on the geometric interpretation of Maxwell’s equations. This technique enables a direct link between a desired electromagnetic (EM) phenomenon and the material response required for its occurrence, providing a powerful and intuitive design tool for the control of EM fields on all length scales. With the unprecedented design flexibility offered by transformation optics (TO), researchers have demonstrated a host of interesting devices, such as invisibility cloaks, field concentrators, and optical illusion devices. Recently, the applications of TO have been extended to the subwavelength scale to study surface plasmon-assisted phenomena, where a general strategy has been suggested to design and study analytically various plasmonic devices and investigate the associated phenomena, such as nonlocal effects, Casimir interactions, and compact dimensions. We review the basic concept of TO and its advances from macroscopic to the nanoscale regimes.

## keywords

## 1 Introduction

Transformation optics (TO) is an emerging technique for the design of advanced electromagnetic (EM) media. It is based on the concept that Maxwell’s equations can be written in a form-invariant manner under coordinate transformations, such that only the permittivity and permeability tensors are modified.^{1}^{–}^{3}

In the past, the form invariance of Maxwell’s equations has been exploited as a computational tool to simplify numerical electrodynamic simulations. In 1996, a transformation from Cartesian to cylindrical coordinates was applied to solve for the modes of an optical fiber with circular cross section.^{1}^{4}^{–}^{8}

## 2 Basic Theory

The physical meaning of coordinate transformation can be given as follows. We start from a Cartesian system with a given set of electric and magnetic fields and their associated Poynting vectors. Next, imagine that the coordinates are continuously distorted into a new system. TO was born of the realization that as the system is distorted it carries with it all the associated fields. Hence, to guide the trajectory of light, only a distortion in the underlying coordinate system is needed, automatically taking with it the light ray. Knowledge of the transformation in turn provides the values of

To give an intuitive view of the TO scheme, we consider a very simple distortion of space: a section of the

Figure 1. A simple coordinate transformation that compresses a space along the

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For a ray propagating along the ^{2}

Note that

To conclude, if we compress a coordinate system along a certain axis, both

It is possible to follow this intuitive approach of compressing and expanding space to the design of much more complex and functional devices. However, leveraging the formal structure of electromagnetism, we can follow a systematic, general approach that allows the consideration of arbitrary transformations. Under a general spatial operation, EM fields are distorted in a way that is exactly equivalent to a transformation of the electric permittivity and magnetic permeability tensors of the form where

#### Table 1. Summary of transformations of different physical quantities.

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#### Table 1. Summary of transformations of different physical quantities.

Before Physical quantities transformation After transformation Scalar potential $\varphi $ ${\varphi}^{\prime}=\varphi $ Charge density $\rho $ ${\rho}^{\prime}=\rho /\mathrm{det}(\overline{\overline{\mathrm{\Lambda}}})$ Electric field $\overline{E}$ $\left[\begin{array}{l}{\overline{E}}^{\prime}\\ {\overline{H}}^{\prime}\\ {\overline{k}}^{\prime}\end{array}\right]={({\overline{\overline{\mathrm{\Lambda}}}}^{\mathrm{T}})}^{-1}\xb7\left[\begin{array}{l}\overline{E}\\ \overline{H}\\ \overline{k}\end{array}\right]$ Magnetic field $\overline{H}$ Wave vector $\overline{k}$ Magnetic flux density $\overline{B}$ $\left[\begin{array}{l}{\overline{B}}^{\prime}\\ {\overline{D}}^{\prime}\\ {\overline{j}}^{\prime}\\ {\overline{S}}^{\prime}\end{array}\right]=\overline{\overline{\mathrm{\Lambda}}}\xb7\left[\begin{array}{l}\overline{B}\\ \overline{D}\\ \overline{j}\\ \overline{S}\end{array}\right]/\mathrm{det}(\overline{\overline{\mathrm{\Lambda}}})$ Electric displacement $\overline{D}$ Current density $\overline{j}$ Poynting vector $\overline{S}$

TO has provided a powerful tool for the design of structures capable of controlling the flow of light. The most well-known example is the EM invisibility cloak proposed in 2006.^{2}^{,}^{10}^{[Fig. 2(b)]}. In this way an external observer would be aware neither of the presence of the cloak nor its contents. In other words, any object hidden in the cloak would be invisible to outer observers [see

Figure 2. (a) The undistorted coordinate system, where a ray of light in free space travels in a straight line. (b) The coordinates are transformed to exclude the cloaked region. Trajectories of rays are pinned to the coordinate mesh and therefore avoid the cloaked region, returning to their original path after passing through the cloak. (c) The coordinates are transformed to fold the space into the annulus region. (d) The field distribution for a cloak under the Gaussian beam illumination. (e) The field distribution for a concentration under the Gaussian beam illumination.

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The corresponding space distortion can be realized through the following coordinate transformation:

This transformation compresses the space in the radial direction. Therefore, along the angular coordinates the values of

If we take a closer look at the cloak parameters, we find that a spherical cloak is singular on its inner surface, i.e., the values of ^{11}^{–}^{14}^{15}^{16}^{–}^{25}

Another example of TO device is the electromagnetic field concentrator that can focus the incident electromagnetic waves to the central area and enhance the electromagnetic energy density.^{26}^{,}^{27}^{28}^{–}^{30}^{31}^{–}^{34}^{35}^{36}^{–}^{38}

## 3 Quasi-Conformal Mapping

To mitigate the material parameter constraints, the flexibility of the coordinate transformations has been explored. Attentions have been turned to the so-called quasi-conformal mapping that allows the design of devices with isotropic dielectric materials or materials with very small anisotropy. As a natural extension from the conformal mapping, the quasi-conformal mapping relaxes the severe restrictions on the conformal mapping while remaining orthogonal such that the permittivity and permeability tensors can be easily realized. These mappings allow for transformations between domains with different conformal modules in two steps. To compensate for the mismatched conformal modules, the virtual domain is first mapped to an intermediate domain with the same conformal module as the physical domain. This can be simply achieved using a uniform compression/expansion

In contrast to conformal mapping, the material tensors produced by quasi-conformal mapping are not equal to each other due to perturbations to the conformal module, as can be seen from Eq. (7). However, these perturbations are generally small and thus the resulting anisotropy can typically be ignored. In general, this technique does not have a closed form analytical solution. However, the quasi-conformal map can be approximated by solving Laplace equation on the coordinates.^{39}

One example of quasi-conformal mapping is the design of “carpet cloak,” which provides an alternative form of invisibility cloak.^{40}^{40}^{41}

Figure 3. A carpet cloak designed with quasi-conformal mapping. (a)

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The material parameters of the carpet cloak are nonsingular and can in principle be realized without the need of resonant features. Furthermore, the isotropic material profile simplifies the fabrication process, making its realization in optical spectrum possible. A number of experimental realizations have been reported, including 2-D and 3-D carpet cloaks working at microwave,^{42}^{–}^{44}^{45}^{–}^{48}^{49}^{,}^{50}

## 4 Linear Transformation

Linear transformation is another type of coordinate transformation used to simplify the device design. In 2-D cases, an arbitrary linear transformation can be described as ^{51}^{,}^{52}

Figure 4. A linear transformation that transforms an arbitrary triangular region to another one in the physical space.

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Therefore, linear transformation opens the way for the design of a homogeneous carpet cloak by linearly compressing the triangular region along the

Figure 5. (a) The linear transformation for the design of a carpet cloak. (b) Full-wave simulation of

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In the transformed geometry, the permittivity tensor of the nonmagnetic carpet cloak takes the form of

Consider a transverse-magnetic (TM, magnetic field perpendicular to the cloak device) polarized Gaussian beam incident obliquely upon such a carpet cloak on top of a flat surface. The anisotropic cloak layer guides the beam around the bump, making the output beam propagate in exactly the same way as that reflected from a flat surface, as shown in

The permittivity tensor in Eq. (10) can be diagonalized by rotating the optical axis, and may be realized with natural birefringent crystals^{53}^{–}^{55}^{56}^{–}^{59}^{53}^{,}^{54}^{56}^{60}^{61}^{62}^{–}^{65}^{34}

## 5 Conformal Transformation

Although quasi-conformal mapping and linear transformation make the realization of devices more feasible, the inhomogeneous or anisotropic distributions of permeability and permittivity are still challenging to achieve, especially at optical spectrum. Conformal transformation is a scheme that can completely eliminate the requisite anisotropy of the material parameters, thus it is especially useful in the device design at optical frequencies.

A conformal mapping is an analytic transformation that preserves local angles. If we consider an analytic function

If we make a coordinate transformation

Moreover, the preservation of local angles ensures that the boundary conditions in the transformed space remain unchanged. Thus, the dielectric constant of each material is also conserved:

Since both the electrostatic potential and the material permittivity are preserved under the 2-D conformal mapping, the delicate design of a metamaterial with a spatial variation in its constitutive parameters is no longer necessary. Thus, conformal transformation not only simplifies the fabrication process but also provides an easy route to engineering the plasmonic properties of the transformed nanostructures. In recent years, optical conformal transformation has been exploited extensively to treat subwavelength fields occurring in plasmonic nanosystems, which are difficult to study analytically with traditional theoretical methods, giving a precise design tool.^{66}^{–}^{69}^{70}^{71}^{72}

## 5.1 Singular Plasmonic Structures

From traditional concepts, it is usually believed that a metallic structure should have a large physical size (as compared to the wavelength) to allow for a broadband light-harvesting process and a nanoparticle of finite size usually sustains localized surface plasmon resonances at discrete, rather than continuous, frequencies. However, there are exceptions to these rules. Some finite nanostructures containing sharp edges (or corners) can behave like infinite plasmonic systems and show a continuous interaction with light over a broad frequency range.^{73}^{,}^{74}

Figure 6. Schematic of the conformal transformation that maps canonical plasmonic systems to singular structures. (a) A thin metal slab that couples to a 2-D line dipole is transformed to a crescent-shaped nanocylinder illuminated by a uniform electric field. (b) Two semi-infinite metal slabs separated by a thin dielectric film that are excited by a 2-D dipole source are transformed to two touching metallic nanowires illuminated by a uniform electric field.

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A thin slab of metal can support surface plasmon excitations with a lower bound cutoff at the zero frequency and an upper bound cutoff at the surface plasmon frequency. However, the energy is dispersed to infinity and cannot be collected for an efficient light harvesting process. By applying a 2-D inverse transformation, which converts the infinite metal slab in the original space into a crescent-shaped cylinder depicted in

Detailed calculations show that absorption cross sections for the two geometries depicted in ^{75}^{76}

As shown in

Figure 7. Absorption cross section as a fraction of the physical cross section as a function of frequency for different shapes of (a) crescents and (b) touching nanowires. The absorption spectrum of one individual cylinder is also shown for comparison. (c) The normalized electric field

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Based on this presented approach, an analytical relationship between a canonical metallodielectric system and a variety of singular plasmonic structures, including wedges, crescents, rough surfaces, touching cylinders, etc., has been established.^{70}^{,}^{75}^{,}^{77}^{–}^{82}

## 5.2 Plasmonic Nanostructures with Blunt Edges/Corners

In real-world applications, singularities or perfectly sharp boundaries in those structures are unlikely to be realized due to limitations in fabrication techniques and the surface tension of the metal. Therefore, the possibility of quantitatively examining how the edge rounding at the sharp boundary will alter the optical responses has great significance on both theoretical and practical levels. TO enables a systematic investigation of a general class of blunt nanostructures by applying conformal mappings to the truncated metallodielectric system associated with the singular structures.^{83}^{,}^{84}^{83}^{85}^{–}^{87}

Figure 8. The original plasmonic systems are truncated periodic metallo-dielectric structures depicted in (a), (c), (e), and (g), where the EM source is an array of line dipoles (red arrows), aligned along the

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Figure 9. Absorption spectrum as a function of the frequency and the bluntness. Figure reprinted with permission: Ref. 83, © 2012 by APS.

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## 5.3 Nonlocal Effects in Plasmonic Nanostructures

In singular structures containing sharp asperities/corners or nearly touching particles, the nonlocal effect, i.e., spatial dispersion in the metal permittivity, plays a key role in the performance of nanodevices, where the classical macroscopic electromagnetism breaks down. An accurate description of optical properties in the subnanometer regime requires the implementation of spatially dispersive permittivities beyond the Drude free electron gas, taking into account the effect of electron–electron interactions. Incorporating nonlocal effects into the TO approach requires the transformation of the permittivity tensor with transverse and longitudinal components under the conformal inversion. The mapping ^{88}^{,}^{89}

Figure 10. Conformal transformation of a metal-vacuum-metal geometry into a nanowire dimer with (a) local^{82}^{88}

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A more general strategy for the inclusion of nonlocal effects in the TO frame can be done through a recently developed simplified model for nonlocality.^{90}^{90}^{91}^{–}^{93}

The nonlocal effects also set an ultimate bound for the maximum field enhancements. As shown in ^{89}

Figure 11. (a) Electric field enhancements in the vicinity of the touching point at different degrees of nonlocality

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## 5.4 Three-Dimensional Structures and Quantum Fluctuation

TO has been extensively employed in 2-D scenarios to investigate the interaction of light with a number of metal structures^{70}^{,}^{75}^{,}^{77}^{–}^{84}^{,}^{88}^{,}^{89}^{,}^{94}^{–}^{96}^{71}^{,}^{99}^{100}

Figure 12. The 3-D inversion that maps a metal-dielectric-metal annulus geometry into a pair of nanospheres separated by a small gap.

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The factor ^{100}^{102}^{–}^{105}^{100}

Figure 13. Resonance frequencies of (a) plasmonic modes and (b) Casimir energy versus the separation between two 5-nm-radius gold spheres. Figure reprinted with permission: Ref. 101, © 2013 by the National Academy of Sciences of the United States of America.

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As mentioned above, the description of the full set of modes enables us to go beyond the optical responses of the structure to study related phenomena in other fields, such as Casimir forces,^{106}^{–}^{108}^{109}^{,}^{110}^{111}^{112}^{101}^{101}^{99}^{95}

## 5.5 Compact Dimension in Singular Plasmonic Metasurfaces

As the last example of this review article, we show how to use the TO approach to compress a whole spatial dimension of a 3-D system into a set of singular points of a 2-D metasurface.^{113}

In traditional optical systems, the number of characteristic ^{113}^{,}^{114}

Figure 14. A conformal transformation compacts (a) the

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To further illustrate the role of the hidden modes, we set

Singular points associated with compact dimensions can be achieved by a number of ways. Apart from the geometric singularities depicted in ^{115}

Figure 15. (a) Schematic of singular graphene metasurface with periodical conductivity. (b) and (c) Absorption spectra of two singular graphene metasurfaces showing how the plasmonic resonances merge into a continuum with increasing dissipation losses. Figure reprinted with permission: Ref. 115, © 2018 by the American Chemical Society.

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## 6 Conclusions

Transformational optics offers unprecedented control over the propagation and confinement of electromagnetic fields at both supra- and subwavelength scales. It not only enables the design of a variety of electromagnetic devices such as the invisibility cloaks and illusion devices but also allows the study of plasmonic nanostructures, revealing the key features that optimize light harvesting and plasmonic field enhancement. Taking into account practical factors including geometric bluntness, nonlocal effects, material limitations, etc., brings the TO approach a step forward toward realistic applications. In fact, the range of applicability of this powerful method is beyond the manipulation of electromagnetic waves and description of the optical properties of metals but can be extended to other physical fields, such as acoustic waves,^{116}^{–}^{118}^{119}^{–}^{121}^{122}^{123}^{,}^{124}

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