## Abstract

Since its inception in 2006, transformation optics has become an established tool to understand and design electromagnetic systems. It provides a geometrical perspective into the properties of light waves without the need for a ray approximation. Most studies have focused on modifying the trajectories of light rays, e.g. beam benders, lenses, invisibility cloaks, etc. In this contribution, we explore transformation optics beyond the manipulation of light trajectories. With a few well-chosen examples, we demonstrate that transformation optics can be used to manipulate electromagnetic fields up to an unprecedented level. In the first example, we introduce an electromagnetic cavity that allows for deep subwavelength confinement of light. The cavity is designed with transformation optics even though the concept of trajectory ceases to have any meaning in a structure as small as this cavity. In the second example, we show that the properties of Cherenkov light emitted in a transformation-optical material can be understood and modified from simple geometric considerations. Finally, we show that optical forces—a quadratic function of the fields—follow the rules of transformation optics too. By applying a folded coordinate transformation to a pair of waveguides, optical forces can be enhanced just as if the waveguides were closer together. With these examples, we open up an entirely new spectrum of devices that can be conceived using transformation optics.

## 1. Introduction

Transformation optics is a framework that exploits the form-invariance of Maxwell's equations in the design of material parameters of optical devices. This form-invariance of Maxwell's equations under coordinate transformations has been discussed by several authors in the past century [1–4] and the equivalence between non-trivial geometries and optical media was already pointed out several decades ago in the study of the propagation of light in exotic gravitational fields [5–7]. Subsequently, Ward & Pendry [8] used this equivalence beyond the scope of gravitational effects to enhance the performance of a finite-elements electromagnetics solver. Recently, Pendry and co-workers conceived the form-invariance of Maxwell's equations from a different point of view, and used it—in analogy with the techniques of optical conformal mapping [9]—as a tool to design novel optical devices within the framework of transformation optics [10]. This design technique has already introduced several new optical devices with unusual properties, such as the invisibility cloak [11–14] and more intricate illusion devices [15,16], advanced lensing systems [17,18] and field manipulators [19,20]. More recently, the technique has also been extended to more general transformations [21], including time-dependent coordinates [22–24], complex coordinates [25] and field transformations [26], which allow for more extensive control over the properties of light in complex metamaterial structures starting from a geometrical perspective.

### (a) Equivalence between geometries and media

In this section, we unveil the form-invariance of the macroscopic Maxwell equations under three-dimensional coordinate transformations [21]. The derivation starts from the macroscopic Maxwell equations in a medium, characterized by the constitutive equations **D**=*ϵ*_{0}*ϵ***E** and **B**=μ_{0}μ**H**. Using the tools of differential geometry, we can expand these vectorial equations in a right-handed Cartesian coordinate system:
1.1where *ρ* and *J*^{i} denote the free charges and currents in the medium. We now transform equations (1.1) to an arbitrary coordinate system, specified by the metric tensor *g*_{ij}. On the background of this metric, we denote the fields and the material tensors with a tilde (∼):
1.2where and [*ijk*] is the permutation symbol, defined by [*ijk*]=1 if (*ijk*) is an even permutation of (123), [*ijk*]=−1 if (*ijk*) is an odd permutation of (123), and [*ijk*]=0 for all other combinations of the triplet (*ijk*).

By comparing equation (1.1) with equation (1.2), we note that the macroscopic Maxwell equations are form-invariant under arbitrary spatial coordinate transformations. It is immediately clear that the effects of a coordinate transformation can be absorbed by the material properties of the medium. Furthermore, this comparison also allows us to identify the relations that govern the equivalence between a coordinate transformation, defined by a metric tensor *g*_{ij}, and the corresponding material implementation. Indeed, we see that the macroscopic Maxwell equations written in an arbitrary coordinate system (equations (1.2)) are identical to Maxwell's equations in the original coordinate system (equations (1.1)) if we assume that the material properties in both sets of equations are related by
1.3and
1.4These relations can be generalized straightforwardly to implement coordinate transformations on the background of a coordinate system that is not Cartesian, but rather characterized by a metric *γ*. In this case, the equivalence relations are given by
1.5and
1.6

### (b) Two different points of view

The previous result embodies the core idea of transformation optics: the correspondence between coordinate transformations and material implementations. We have thus demonstrated the equivalence between two different spaces—two different points of view: the *electromagnetic space*, which usually contains trivial materials (vacuum or isotropic dielectrics) and which is characterized by the metric components *g*_{ij}, and the *physical space*, which contains a metamaterial whose properties are determined by equations (1.5) and (1.6), and is expressed on the background of a right-handed Cartesian coordinate system [21]. Although the metric components of electromagnetic space *g*_{ij} can refer to a general curved geometry with arbitrary coordinates, we will usually consider a flat manifold as background, such that the non-trivial *g*_{ij} components are merely the result of expressing flat space–time in an unusual coordinate system.

The relevance of this equivalence between coordinate transformations and media is twofold. First, it is important from a theoretical point of view, as it offers a new perspective to understand the behaviour of light in macroscopic media. Indeed, using the equivalence relation of transformation optics, light propagation through a complex macroscopic medium can be transformed to light propagation through a trivial medium, e.g. vacuum, expressed in an unconventional coordinate system. Second, this equivalence provides an effective tool to design electromagnetic components from a geometric perspective. The procedure is shown in figure 1, where we show the consecutive steps in the design of optical components using transformation optics. There is, off course, an intricate cross-fertilization between both perspectives: using a geometric formalism to obtain an increased understanding in the propagation of light through complex dielectrics often leads to novel methods for the design of optical components. Such interplay is, e.g. highlighted in §3, where we investigate the emission of Cherenkov radiation in complex metamaterials.

The design procedure usually starts with a space in which the propagation of light is well understood, e.g. vacuum or a simple dielectric, expressed in a traditional (Cartesian) coordinate system (figure 1*a*). In the next step, we transform this electromagnetic problem to a new, deformed coordinate system, identified by the metric tensor *g*_{ij} (figure 1*b*). Finally, the equivalence relations of transformation optics, equations (1.5) and (1.6), allow for the calculation of the material properties *ϵ* and μ—and in some cases *ρ* and **J**—that reproduce the same electromagnetic fields as the ones from the curved electromagnetic space in the original (Cartesian) coordinate system (figure 1*c*).

The design method bears some similarities to the optical conformal mapping technique [9]. One could say that transformation optics provides a generalization of this technique in a number of different aspects.

First and foremost, the equivalence relations of transformation optics are valid for the full-wave Maxwell equations. Indeed, in contrast to optical conformal mapping, transformation optics successfully incorporates geometrical concepts to manipulate the wavelike behaviour of light, a regime that is known to be incompatible with the geometrical concepts used in the eikonal approximation. This can be understood from the fact that conformal transformations transform only the Helmholtz equation with form invariance. In the case of media with inhomogeneous permittivity and permeability, however, one needs to make a short-wavelength approximation to put the wave equation for the electromagnetic fields in the Helmholtz form. Variations of the refractive index profile will thus introduce unwanted distortions in the propagation expected from the conformal map. Transformation optics overcomes this approximation by transforming the full Maxwell equations.

This is why it is somewhat misleading to denominate the red line in figure 1 as the ray trajectory of light. It is more correct to identify the line with the flow of electromagnetic energy, i.e. Poynting's vector. In this manuscript, we highlight this undervalued aspect of transformation optics by investigating applications whose functionality goes beyond the ray approximation. The miniaturization of optical resonators, discussed in §2, and the enhancement of optical gradient forces, discussed in §4, are two examples where we apply transformation optics in a deeply subwavelength regime.

A second generalization of transformation optics with respect to optical conformal mapping is the introduction of *anisotropy* in the material implementation of general coordinate transformations. Isotropic material implementations correspond to conformal maps.

Third, transformation optics allows for *local transformations* of the coordinate system. It is possible to manipulate light within a certain finite region using a non-trivial coordinate transformation that approaches unity at the boundaries of the region. In this way, it is possible to create a finite component that smoothly manipulates light without reflections at the boundaries. This aspect of transformation optics is closely related to the impedance matching of the equivalence relations: coordinate transformations affect the permittivity and the permeability of electromagnetic space in the same way. The impedance of the optical component in physical space will thus be the same as in the initial system in electromagnetic space.

In the following sections, we review our original research contributions to the aforementioned fields. Each section is focused on a specific application or extension of the methodology of transformation optics. In §2, we introduce transformation optics as a novel tool to design optical cavities. Indeed, we show that the invisibility cloak can be inverted to prevent light from escaping a finite region. Using this technique, we demonstrate that it is possible to eliminate bending losses, which are normally limiting the quality factor of optical resonators. Subsequently, in §3, we apply transformation optics to calculate the Cherenkov cone emitted by charged particles travelling through anisotropic media. We show that Cherenkov radiation obeys the geometry of the electromagnetic reality. The Cherenkov cone can thus be manipulated with material parameters that implement coordinate transformations. Finally, in §4, we investigate linear momentum transfer between light and matter from the perspective of transformation optics. This is becoming an important physical mechanism for all-optical actuation in micro- and nanophotonic systems. We investigate the typical set-up of two coupled slab waveguides and discuss how optical gradient forces depend on the waveguide's parameters. We find that the optical gradient forces perfectly coincide with the forces in the underlying electromagnetic space when complex waveguide materials are considered. Based on this observation, we present a novel mechanism to enhance optical gradient forces between waveguides using a cladding of single-negative metamaterials.

## 2. Transforming space to confine light

### (a) Dielectric microresonators

The possibility to confine light in small volumes is a crucial element both in theoretical studies of the quantum mechanical behaviour of light [27,28] and in the realization of advanced information processing systems [29,30]. Traditionally, light confinement is achieved using different types of optical cavities, such as Fabry–Pérot cavities, photonic crystal cavities and dielectric microcavities [31]. These dielectric microcavities consist of a purified dielectric material with a high index of refraction *n*_{in} surrounded by another dielectric with a lower index of refraction *n*_{out}, e.g. air. Light rays can then be trapped inside a cylindrical or spherical geometry due to internal reflection at the boundaries. This mechanism is shown in figure 2. Resonant confinement occurs when light constructively interferes with itself after one round-trip.

This geometrical interpretation is a rather crude approximation, specifically when the resonator's dimensions are reduced to the same magnitude as the wavelength of confined light. In this mesoscopic regime, effects of diffraction, which are neglected in the ray picture, should be taken into account. The confinement is thus partial because some energy will always be lost to the surrounding environment. Therefore, the eigenmodes of these optical cavities—so-called quasi-normal modes—are characterized by a discrete set of complex eigenfrequencies, where the real part (*ω*′) is proportional to the inverse of the wavelength of the confined light and the imaginary part (*ω*^{′′}) is a measure of the temporal confinement of the wave inside the cavity.

### (b) Dispersion relation of a cylindrical cavity with a radial coordinate transformation

In the following subsections, we will review several new approaches to design optical cavities within the framework of transformation optics. In order to do so, we first derive the dispersion relation of a cavity based on a radial coordinate transformation.

The system under investigation, shown in figure 3, consists of an infinite cylindrical cloak that is bound by radii *ρ*=*R*_{1} and *ρ*=*R*_{2}. Regions (I) and (III) are vacuum and region (II) contains the medium that performs the coordinate transformation. Without loss of generality, we will consider the time-harmonic solutions that are polarized along the *z*-axis: .

We will look for the eigenmodes of this system by solving Maxwell's equations inside each region and matching the solutions using the proper boundary conditions. Inside the vacuum regions, the solutions of Maxwell's equations are well known: the angular variation of the electric field is a sum of complex exponentials, characterized by indices *ν*_{I} and *ν*_{III}, whereas the radial dependence obeys the Bessel equation. The radial dependence of the electric field can therefore be written as a sum of Bessel functions *J*_{ν} and *Y*_{ν} or as a sum of Hankel functions and . In the inner region (I), we decompose the field using the Bessel functions where we can reject the Bessel function *Y*_{ν}, because it has a singularity at the origin. In the surrounding vacuum region (III), we decompose using the Hankel functions, where we can drop the Hankel function , which represents an incoming wave:
2.1and
2.2where *A*_{νI} and *C*_{νIII} are complex integration constants and *k*_{0}=*ω*/*c* represents the vacuum wave number. In order to calculate the solutions in the transformation-optical region (II), we need to insert the constitutive parameters for the material in Maxwell's equations. These constitutive equations can be derived from the coordinate transformation, using the equivalence relations of transformation optics, equation (1.6). In this section, we consider the case of an arbitrary radial transformation, leaving the azimuthal angle and the *z*-axis unchanged: (*ρ*,*ϕ*,*z*) is transformed into (*ρ*′,*ϕ*′,*z*′) such that *ρ*′=*f*(*ρ*), *ϕ*′=*ϕ* and *z*′=*z*. Transformation optics now tells us that such a distortion of the radial coordinate in vacuum is analogous to a medium with the following components of the permittivity and permeability tensors:
2.3where the prime denotes differentiation with respect to *ρ*. By insertion of these parameters in Maxwell's equations, it can be shown that the solutions inside the transformation-optical region (II) are given by [32]
2.4where, once again, *F*_{νII} and *G*_{νII} are complex integration constants.

This solution needs to be matched to the solutions of the electromagnetic field in regions (I) and (III) using the Maxwell boundary conditions. The dispersion relation is thus retrieved by imposing the continuity of the tangential components of the electric (*E*^{z}) and magnetic fields (*H*^{ϕ}) at the boundaries between the regions. Because of the symmetry in the angular direction, the azimuthal mode number *m*_{i} remains the same in each region (*m*_{i}=*m*) and, as a result, we find the following set of equations that govern the eigenmodes of the cavity:
2.5
2.6
2.7
2.8where *A*_{m}, *F*_{m}, *G*_{m} and *C*_{m} are complex integration constants. It is important to note that this relation is valid for any cavity of the type shown in figure 3, implementing a radial coordinate transformation *ρ*′=*f*(*ρ*) between *R*_{1} and *R*_{2}. In the following sections, we will evaluate this relation to discuss the confined modes of two cavities, whose functionality is based on a radial coordinate transformation.

### (c) Making the electromagnetic space larger

A cavity ideally confines electromagnetic energy in small region of space for a very long time. This can be interpreted in terms of coordinate transformations, by mapping a large domain of electromagnetic space onto a much smaller region in physical space. An example of such a transformation is a hyperbolic map, which grows to infinity at a finite point. Therefore, we can consider a coordinate transformation, in which , where
2.9and
2.10The transformed Cartesian coordinate lines are shown in figure 4*a*. The matching at the inner boundary enables a smooth transition of the waves. As there cannot be anything ‘beyond infinity’, equation (2.10) ensures that electromagnetic energy cannot escape the device. In figure 4*a*, where we consider the transformation given by *f*(*ρ*)=*R*_{1}(*R*_{1}−*R*_{2})/(*ρ*−*R*_{2}), the Cartesian coordinate lines in physical space become denser as we approach the outer radius *R*_{2}. The values of the material parameters required to implement this transformation in physical space are shown in figure 4*b*.

The determinant of equations (2.5)–(2.8) can be calculated in the limit of , using equation (2.9). We find that the hyperbolic map does not confine any electromagnetic modes. We find that, independent of the azimuthal mode number *m*, the dispersion relation can only be satisfied if *k*_{0}=0, i.e. the static solution. This result fits with the intuitive idea that in this configuration an electromagnetic wave travels an infinitely long time to reach the outer boundary of the cavity. Therefore, no standing wave can be created in the cavity: waves are not reflected at the outer boundary, *ρ*=*R*_{2}.

### (d) Folding the electromagnetic space

An invisibility cloak turns a finite region in space invisible by guiding light around it. For an ideal cloak, it is impossible for light of any wavelength to enter the cloaked region. A perfect invisibility cloak is characterized by a continuous radial coordinate transformation for which the inner boundary in the physical space is mapped onto the origin in electromagnetic space, whereas the outer boundary is mapped onto itself [10]:
2.11and
2.12We can, however, construct a cavity from this cloaking perspective and design a device that cloaks away the volume surrounding the device instead of the volume inside the device [33]. Such a device should smoothly guide the electromagnetic waves in the cavity so that they never penetrate the outer boundary. The effect of such a transformation is shown in figure 5*a*. As we want to cloak away region (III), we will use a radial coordinate transformation that maps the physical coordinates (*ρ*,*ϕ*,*z*) onto the electromagnetic coordinates (*ρ*′,*ϕ*′,*z*′). To achieve perfect cloaking of region (III) from the viewpoint of region (*I*), the radial transformation function has to satisfy the following boundary requirements:
2.13and
2.14

To calculate the modes of this cavity, we need to evaluate equations (2.5)–(2.8), where we now have to insert *f*(*R*_{1})=*R*_{1} and *f*(*R*_{2})=0. In this case, we find
2.15
2.16
2.17
2.18These limits should be handled carefully, as they contain infinities and indefinite expressions. For azimuthal mode numbers *m*≠0, we can evaluate these limits unambiguously ( and ), because of which the coefficient *G*_{m} needs to be zero. The previous set of equations then transforms into
2.19and
2.20Surprisingly, there is no quantization of the wavevector *k*_{0}. In other words, the cavity supports a continuous spectrum of modes, even wavelengths that are much larger than the size of the cavity can be confined. Furthermore, the quality factor of these modes is infinite: there is no tunnelling of electromagnetic energy into the surrounding vacuum.

### (e) Discussion

We thus demonstrated two different mechanisms to trap light using transformation optics. Physically, both mechanisms are based on the fact that a wave can only be confined inside a cavity if one round-trip (approximately the cavity's circumference) equals an integer number *l* of the mode's wavelength inside the cavity: 2*πa*≈*lλ* [34]. The hyperbolic map, on the one hand, reduces the wavelength of the electromagnetic wave such that there are no reflected waves and there is no round-trip. In the case of the folding coordinate transformation, on the other hand, the phase shift vanishes completely and *l*=0.

It should be noted that the parameters required to materialize these coordinate transformations are difficult to achieve in a practical set-up. In particular, the zeros and the infinities at the outer boundary of the cavities cannot be strictly implemented. Therefore, we have also performed additional numerical simulations in which realistic metamaterial implementations are investigated [33]. Qualitatively, the same geometrical results remain valid in these simulations. The dominant effect of these non-ideal material parameters is the reduction of the overall quality factor of the modes, which is then entirely determined by the loss tangent of the materials under consideration. These losses are large inside materials with a refractive index close to zero. This can be understood as follows. In regions where the refractive index approaches zero, the phase velocity of the electromagnetic fields (*v*_{ϕ}=*c*/*n*) approaches infinity. This means that the group velocity *v*_{g}—bounded by the speed of light in vacuum—should significantly differ from the phase velocity *v*_{ϕ} which in turn implies significant dispersion. Therefore, following Kramers–Kronig relations, these regions are usually dissipative, albeit not necessarily at the same frequency. However, using an azimuthal transformation in addition to the radial folding transformation, it is possible to arrive at subwavelength transformation-optical cavities made from purely right-handed media. This substantially eases the material implementation. In addition, one can remove the singular rim at the outer boundary, containing the highly dispersive materials—it is then still possible to arrive at subwavelength cavities with small (but not vanishing) radiation losses [32,33].

## 3. Transforming space to manipulate the emission of light

### (a) Cherenkov radiation

Cherenkov radiation [35–37] is a peculiar form of electromagnetic radiation that arises when charged particles travel through a medium at a velocity greater than the phase velocity of light in that medium [38], as demonstrated in figure 6. Experimentally, this radiation was discovered by Pavel Cherenkov [39] and, theoretically, it was formalized by Frank & Tamm [40]. Nowadays, this effect is well understood and has been proved useful in a wide range of applications in applied and experimental physics [41], including high-energy particle physics, detection of cosmic rays in astrophysical measurements [42], development of novel electromagnetic sources [43–45], localized sensing in biological systems [46], spectroscopy of complex nanostructures [47].

In recent years, there has been significant interest in the manipulation of Cherenkov radiation inside or in the vicinity of electromagnetic structured media [48–61]. The interest generated from the prediction that, under certain conditions, the direction of the cone of Cherenkov radiation can be reversed, such that the Cherenkov radiation and the emitting particle travel in opposite directions [62]. Aside from this phenomenon—called reversed Cherenkov radiation [48–53,56,57]—other unusual phenomena, such as enhanced Cherenkov radiation or Cherenkov radiation without velocity threshold [60] have been predicted when charged particles travel through metamaterials and other electromagnetic structured systems. In this section, we will generalize these findings by investigating Cherenkov radiation using the methodology of transformation optics [63].

### (b) Cherenkov emission in a transformation-optical medium

#### (i) In physical space

We start by calculating the Cherenkov radiation that is emitted by a moving charge, propagating along the *x*-axis of a medium with material parameters that correspond to a background refractive index (*ϵ*_{b}=*n*^{2}_{b}) on top of which a linear coordinate stretching along the principle axes has been implemented: *x*′=*f*(*x*), *y*′=*g*(*y*) and *z*′=*h*(*z*). Following the equivalence relation of transformation optics, the material parameters of this medium are given by
3.1The Cherenkov radiation can be calculated most conveniently by way of the electromagnetic potentials in the generalized Lorentz gauge:
3.2The inhomogeneous Maxwell equations can then be cast in the following form:
3.3and
3.4where we introduced the reluctance tensor as the inverse of the permeability tensor . We can investigate these equations for a fixed value of **k** in momentum space:
3.5and
3.6from which **V**_{k} and **A**_{k} can be calculated. These expressions imply that the electromagnetic field with wavevector **k** generated by a charged particle, travelling at a velocity **v**, will oscillate at a frequency *ω*=**k**⋅**v**. For a given frequency *ω*, the direction of the emitted wave is obtained by matching *ω*=**k**⋅**v** with the dispersion relation of the medium in which the charged particle is moving [38].

The dispersion relation of the medium, defined by equation (3.1), can be found by inserting a plane monochromatic wave solution **E**=(*E*_{x}**1**_{x}+*E*_{y}**1**_{y}+*E*_{z}**1**_{z}) exp[i(**k**⋅**r**−*ωt*)] in the wave equation
3.7This results in the following set of equations:
which has a non-trivial solution if its determinant equals zero. This yields the following dispersion relation:
3.8

We can now apply this dispersion relation to calculate the angle of Cherenkov radiation in a transformation-optical medium. Without loss of generality, we can restrict this analysis to the *xy* plane. Writing *f*′(*x*)=*F*, *g*′(*y*)=*G*, *h*′(*z*)=*H* and defining *α*_{PH} as the angle under which the electromagnetic waves are emitted (figure 7), we find that
3.9where *α** is the angle of Cherenkov radiation emitted in a medium with refractive index *n*_{b}*F*. This angle is simply given by the traditional Cherenkov formula and, hence, we find that
3.10

### (c) In electromagnetic space

The previous result can be better understood from a transformation-optical perspective. In physical space, the charge is moving at velocity *v* in the *x* direction. As this coordinate is stretched by a factor *F*, the particle seems to be moving at velocity *Fv* in the underlying electromagnetic space. In this space, the particle simply travels through an isotropic medium with refractive index *n*_{b} and, therefore, emits Cherenkov radiation with an opening cone *θ*_{EM} given by
3.11Translating the emitted radiation back to the physical space, the *x* and *y* components of the cone need to be compressed by a factor *F* and *G*, respectively, in the underlying electromagnetic space. The angle of the Cherenkov cone then becomes
3.12Obviously, an analogous relation is valid in the *xz* plane, whose angle can be stretched by
3.13

### (d) Discussion

It is clear that a transformation of a coordinate in the longitudinal direction has a fundamentally different effect on the Cherenkov radiation than a transformation of a coordinate perpendicular to the direction of propagation, as can be seen in figure 8. This is related to the fact that a transformation perpendicular to the trajectory of the charged particle only stretches the Cherenkov cone, whereas a transformation along the path of the charge also alters the velocity of the charge in the underlying electromagnetic space. As soon as this velocity drops below the speed of light *c*/*n*_{b}, Cherenkov radiation ceases to exist. This Cherenkov cut-off is visible in figure 8*b*, when the longitudinal scaling factor *F* approaches *c*/(*n*_{b}*v*)=0.5. In this respect, a longitudinal coordinate transformation has a remarkable effect. Consider, for example, a charged particle travelling through a dielectric (with refractive index *n*_{b}) at a velocity *v* smaller than the phase velocity of light in that medium *v*<*v*_{ϕ}=*c*/*n*_{b}. Subsequently, it is possible to change to a novel coordinate system, in which the particle seems to be traveling faster than the speed of light in the dielectric. This is a consequence of a longitudinal transformation that can scale the particle's velocity above or below the Cherenkov cut-off velocity. To implement this transformed space inside a specific material, one needs to change the refractive index of the dielectric (*n*=*n*_{b}*F*, where *F* is a scaling factor determined by transformation optics), because of which the phase velocity of light also changes in this medium. In this new medium, it is then clear that *v*>*v*_{ϕ}=*c*/*n*, which explains the existence of Cherenkov radiation in the transformed medium.

## 4. Transforming space to enhance optical gradient forces

In this section, we generalize the scope of transformation optics to include the study of momentum transfer in complex metamaterial structures. The momentum of light has been a fascinating subject for scientists for several centuries [64–66] and even nowadays, the true nature of the photon's momentum remains disputed [67,68]. At macroscopic scales, this momentum is normally too small to have any significant effect, but in nanoscale devices the transfer of linear momentum between light and matter and the associated optical forces start to play an increasingly important role [69].

Generally, these forces can be divided into scattering and gradient forces, depending on whether the transferred momentum is parallel or perpendicular to the direction of propagation. Optical scattering forces have been used to cool down atoms through the interaction with intense laser light [70–72] and, more recently, for the generation of tractor beams [73] and in the field of cavity optomechanics where the coupling between the optical and mechanical modes of a cavity is exploited to manipulate the vibrations of a mechanical system [74,75]. Optical gradient forces are used in optical tweezers, where microscopic dielectric particles are trapped and moved by laser beams towards regions of highest intensity [76].

In recent years, there is increasing interest to calculate optical forces in complex artificial structures [69,77–85]. Most recently, scientists discussed the use of optical forces to manipulate metamaterials on a microscopic level [78,84,85]. Optical forces indeed offer an elegant approach to achieve scalable nonlinearities in metamaterials. It was first demonstrated by Zhao *et al.* [78] that there is a very rich behaviour of optical forces between two patterned metallic sheets, for example sheets of nanowire pairs. Similar behaviour was demonstrated by Zhang *et al.* between a metallic and a dielectric sheet [84]. These optical forces can also be taken advantage of for nonlinear magneto-elastic metamaterials, as was shown by Lapine *et al.* [85]. They demonstrated that a scalable nonlinearity can be obtained by coupling the electromagnetic interaction to the elastic interaction in a metamaterial lattice by considering an anisotropic lattice of resonant split-ring resonators. For a magnetic field oriented perpendicularly to the plane of the circular split-ring resonators, a current flows inside the resonators. The currents induced in all the resonators produce a force between the resonators, giving rise to a nonlinearity in the effective medium parameters of the metamaterial.

### (a) Optical gradient forces in coupled waveguides

#### (i) Method

We now consider optical gradient forces between coupled dielectric waveguides. To this aim, we calculate the forces that act on two slab waveguides, as shown in figure 9. To evaluate the forces that act on the rightmost waveguide, we need to evaluate the fields in regions I, II and III. Analogously to the derivation of quasi-normal modes inside dielectric microresonators, the waveguide modes are found by solving Maxwell's equations in combination with appropriate boundary conditions. However, in this section, we are looking for the propagating modes of the waveguide system and, therefore, we use the ansatz **E**=*E*(*x*) exp[ i(*βy*−*ωt*)]**1**_{z}.

The mode profile *E*(*x*) is a solution of a Helmholtz equation in each region (I, II and III):
4.1where *n*_{i}=*n* inside the dielectric core of the waveguide and *n*_{i}=1 in the vacuum surroundings. The requirement that the field vanishes at infinity implies that and .

A general solution of the electromagnetic fields in this symmetric waveguide set-up can always be decomposed as a sum of even and odd solutions. Therefore, we can restrict our analysis to the even or odd modes. In the case of even modes, we obtain the following solutions for the fields in each region
4.2
4.3
4.4
4.5
4.6where and . The odd mode solution is similar, apart from the hyperbolic cosine being replaced by a hyperbolic sine in region I and the signs in front of *B*, *C* and *D* being inverted in regions IV and V.

The dispersion relation of this coupled-waveguides system can be obtained using the requirement that the tangential components of both the electric and magnetic fields are continuous at each boundary. This eigenvalue problem determines the solutions *β* as a function of the frequency *ω*, where the corresponding eigenvector is the set of amplitudes (*A*, *B*, *C* and *D*) of the eigenmode. The resulting electric field profile of a typical even mode solution is shown in figure 9.

We now proceed by calculating the optical forces that act upon the rightmost waveguide of figure 9. Application of the Maxwell stress tensor formalism to the test volume *V* yields
4.7
4.8The integrals over *S*_{3} and *S*_{4} cancel, because of invariance in the *z*-direction. Consequently, we only need to calculate the integrals over *S*_{1} and *S*_{2}. This is a very powerful aspect of the Maxwell stress tensor formalism. We do not need to know the actual field distribution *inside the component*, we only require the fields surrounding the component to calculate the forces that act upon it. Let us first evaluate the surface integral over *S*_{1}:
4.9where we applied the time-averaging product rule, with *B** the complex conjugate of *B*. We continue by using equation (4.2) in combination with Faraday's law and retrieve
4.10Inserting , we end up with
4.11where Δ*S*_{1} is the area of the surface *S*_{1}. We note that this result does not depend on the position of the surface *S*_{1}—the variable *x* drops out of the final result, as it should be. Otherwise, there would exist a vacuum contribution to the overall optical force. Analogously, we can calculate the surface integral over *S*_{2}:
4.12which vanishes to zero after insertion of . Finally, we find that the time-averaged force per unit area on each of the waveguides equals
4.13

#### (ii) Results

We evaluate equation (4.13) to calculate the optical gradient force as a function of the parameters of the waveguides. In particular, we investigate the dependence on the gap distance, the waveguide thickness and the refractive index of the core. Clearly, the optical force is proportional to the amount of optical power that is coupled into the waveguide system. Therefore, we normalize the field amplitude in the following simulations to an input power of 1 mW μm^{−1}.

In figure 10*a*, we plot the optical force between two coupled slab waveguides as a function of their separation distance. The highest forces are obtained for the smallest separation distances, which is in agreement with the fact that the interaction between one waveguide and the evanescent tails of the other waveguide is higher at small distances, as shown in figure 10*b*,*c*. The force *decays exponentially with increasing gap distances*.

#### (iii) Discussion

The magnitude of the gradient forces between two slab waveguides depends strongly on the geometric parameters and refractive indices of the waveguides (figures 10–12). In general, the magnitude increases as the interaction between the evanescent tails and the waveguides becomes stronger. The interaction strength is largest when the separation distance and the waveguide thickness are small. Therefore, realistic set-ups, that aim for measurable displacements, typically use very thin silicon waveguides (*w*=±300 nm) that are positioned very close to each other (gap=±100 nm) [86,87]. The displacement of the waveguides can further be enhanced by reduction of the mass of the waveguides and enlargement of the waveguide length.

Nevertheless, the resulting displacements remain too small to be useful for all-optical actuation. Therefore, several techniques have been developed to enhance the gradient forces inside integrated systems. Typically, the approach consists of enhancing the optical fields at certain positions. Hence, these methods allow for amplification of the optical force per milliwatt of *input* power, although—locally—the force per milliwatt of optical power remains the same. We mention the so-called resonator-based enhancement and the slow-light enhancement techniques as two examples [88,89].

### (b) Transforming optical gradient forces

Here, we show how transformation optics can be used to manipulate optical gradient forces. We start by demonstrating that the methodology of transformation optics is applicable to investigate and design electromagnetic momentum transfer in complex metamaterial structures. To this aim, we have performed a set of simulations in which we calculated the optical forces that arise in coupled slab waveguides, whose material parameters are modulated in agreement with a specific coordinate transformation. The results of these simulations are shown in figure 13. In figure 13*a*, we show the optical force between to traditional slab waveguides (*n*=2.5) as a function of the separation distance for several values of the waveguide thickness between 100 and 600 nm. In figure 13*b*, on the other hand, we show the force that arises between two waveguides with fixed thickness *t*=200 nm on top of which a stretching/compression of the *x*-coordinate, *x*′=*f*(*x*), has been implemented through an additional anisotropy of the permittivity and the permeability. The coordinate transformation is chosen such that the electromagnetic thickness *t*' of the waveguides is the same as the thickness of the waveguides studied in figure 13*a*. The material parameters of the waveguides in figure 13*b* are then given by
4.14where *n* is the refractive index of the waveguides in electromagnetic space.

Although *the actual electromagnetic field profiles in physical and electromagnetic space are unequal*, we find excellent agreement between the forces in both sets of simulations. The reason is that the forces are entirely determined by *the field profile between both waveguides* (region I). In this region, the fields are indeed exactly the same in physical and electromagnetic space. This result demonstrates that optical forces comply with the underlying electromagnetic geometry of a specific system.

### (c) Enhancing optical gradient forces

We now review our mechanism to enhance optical gradient forces using the method of transformation optics [90]. The idea originates from the observation that the optical gradient force between two waveguides decays exponentially with the distance between the waveguides. Transformation optics allows to engineer the interwaveguide distance perceived by light. We achieve this result by designing a medium that annihilates the optical space between two objects by implementing a folded coordinate transformation. We study two waveguides separated by such an annihilating optical medium. For symmetry purposes and practical feasibility, the metamaterial cladding can be attached to the interior boundaries of both waveguides.

To annihilate a distance that is *α* times larger than the physical thickness *t* of the metamaterial cladding, we need a transformation that maps the physical coordinate *x* to the electromagnetic coordinate *f*(*x*)=−*αx*. This can be understood from figure 14, from which it is clear that the two physical points *x*_{1} and *x*_{3} are mapped on the same point *x** in electromagnetic space. Consequently, the electromagnetic fields are identical in points *x*_{1} and *x*_{3}. The position of *x*_{1} depends on the transformation inside the metamaterial slab: *x*_{1}=*x*_{2}−*α*(*x*_{3}−*x*_{2})=*x*_{2}−*αt*. Inserting the function *f*(*x*) in the equivalence relation of transformation optics results in the following materials parameters: *ϵ*_{xx}, μ_{xx}=−1/*α*, *ϵ*_{yy}, μ_{yy}=−*α*, and *ϵ*_{zz}, μ_{zz}=−*α*, where we used the linear transformation *f*(*x*)=−*αx*.

### (d) Discussion

Our study shows that transformation optics allows for the design of materials which can significantly enhance the optical gradient force between two slab waveguides—the prototype system for the study of optical gradient forces. When the space-annihilating waveguide cladding is implemented by a double-negative metamaterial, the force amplification is limited by dissipative loss.

However, we can also use a single-negative metamaterial as a ‘poor man's’ version of the annihilating medium [90]. This approach results in much larger force enhancement factors while not impacting the mechanical properties of the waveguide system. Indeed, the use of single-negative metamaterials, which can be implemented with a stack of thin metal sheets, has the added advantage of low dissipative loss and the mechanical properties will not be adversely affected by the metamaterial structure. Furthermore, this implementation is remarkably robust to the dissipative loss normally observed in metamaterials.

## 5. Conclusion

Transformation optics sheds new light on the propagation of light in complex media. Based on the equivalence between curved geometries and inhomogeneous electromagnetic materials, it allows for the investigation of the propagation of light using the tools of differential geometry. In this way, transformation optics is incorporating geometrical intuition in the design of electromagnetic components, with accuracy beyond the geometrical approximation. The elegance of the approach lies in the intrinsic abstraction of the parameters of the incident light, such as the polarization, the frequency or the propagation direction.

The research in transformation optics is highly focused on the development of devices that allow for advanced manipulation of ray trajectories, the primary example being the invisibility cloak. In our opinion, several interesting aspects of transformation optics remain unexplored in the scientific literature. We have addressed some of these issues and we deliberately searched for applications where the methodology of transformation optics could introduce major advances.

First, we think that one of the main advantages of transformation optics in comparison with the traditional geometrical techniques is its ability to design optical components with subwavelength dimensions. Moulding the flow of light in subwavelength dimensions is particularly interesting in the study of electromagnetic confinement. Indeed, traditional electromagnetic cavities cannot be miniaturized because of diffraction, which severely limits the use of optical resonators for research and industrial applications. Therefore, we introduced the idea of using transformation optics to confine light in electromagnetic resonators. We showed that the dispersion relation of a cavity that materializes a coordinate transformation coincides with *the dispersion relation of the underlying electromagnetic space*. As a result, the implementation of a coordinate transformation inside a cavity structure allows for *imposing different boundary conditions* at the physical edges of the resonator. Based on these insights, we demonstrated the design of several cavities with unconventional characteristics. Most notably, we presented a perfect cavity, which is a resonator in which light of any frequency can be confined without the appearance of bending losses.

Second, apart from work in antenna design, little interest was shown in the study of light generation using transformation optics. We investigated the electromagnetic radiation that is emitted by charged particles, travelling at a constant velocity through a transformation-optical medium. Here as well, we demonstrated that the resulting *Cherenkov radiation can be understood from the propagation of the particle in the underlying electromagnetic reality*.

In addition, we have investigated whether the formalism of transformation optics can be extended to incorporate momentum transfer between light and matter. This physical mechanism and the associated optical forces become an increasingly important tool to actuate nanoscale devices with light. One of the main results in this topic is our discovery that *optical forces between complex dielectrics comply with the underlying electromagnetic space*. Consequently, we have shown that the forces that arise between dielectrics can be enhanced significantly using a folded coordinate transformation. It is intriguing to note that metamaterials—which were initially developed to manipulate light with matter—can thus also be employed to manipulate matter with light.

## Authors' contributions

V.G. and P.T. have designed the study, carried out the experiments and drafted the manuscript.

## Competing interests

The authors declare that they have no competing interests.

## Funding

Work at the Vrije Universiteit Brussel was supported by BelSPO (grant IAP P7-35, photonics{at}be). V. G. is a Postdoctoral Research Fellow of the Research Foundation-Flanders (FWO-Vlaanderen).

## Footnotes

One contribution of 14 to a Theo Murphy meeting issue ‘Spatial transformations: from fundamentals to applications’.

- Accepted April 7, 2015.

- © 2015 The Author(s) Published by the Royal Society. All rights reserved.