Spatial transformation-enabled electromagnetic devices: from radio frequencies to optical wavelengths

Abstract

Transformation optics provides scientists and engineers with a new powerful design paradigm to manipulate the flow of electromagnetic waves in a user-defined manner and with unprecedented flexibility, by controlling the spatial distribution of the electromagnetic properties of a medium. Using this approach, over the past decade, various previously undiscovered physical wave phenomena have been revealed and novel electromagnetic devices have been demonstrated throughout the electromagnetic spectrum. In this paper, we present versatile theoretical and experimental investigations on designing transformation optics-enabled devices for shaping electromagnetic wave radiation and guidance, at both radio frequencies and optical wavelengths. Different from conventional coordinate transformations, more advanced and versatile coordinate transformations are exploited here to benefit diverse applications, thereby providing expanded design flexibility, enhanced device performance, as well as reduced implementation complexity. These design examples demonstrate the comprehensive capability of transformation optics in controlling electromagnetic waves, while the associated novel devices will open up new paths towards future integrated electromagnetic component synthesis and design, from microwave to optical spectral regimes.

1. Introduction

The majority of electromagnetic devices have conventionally been designed by a feed-forward procedure, which involves the realization of certain functionalities from the available materials. Such a procedure greatly limits the rich possibilities of wave–matter interactions, making the discovery process of the best solution difficult and time consuming. In 2006, transformation optics, a new mathematical tool based on the form invariant properties of Maxwell's equations, was proposed by Leonhardt [1] and Pendry et al. [2]. This new approach provides a systematic inverse design paradigm to control the trajectories of electromagnetic waves by deducing the spatial distribution of material properties from a designed coordinate transformation, as required by the desired functionality [3–5]. The transformation optics methodology has ushered in a new era of engineering wave–matter interaction, which enables a comprehensive and delicate control over the propagation, radiation, scattering and guidance of all forms of electromagnetic waves [6].

Ever since the pioneering theoretical work reported in [1,2], the field of transformation optics has witnessed a dramatic growth, leading to a wide variety of devices that achieve novel electromagnetic functionalities. Two of the most interesting and exotic examples are the invisibility cloaks and illusion devices. The cloak is an inhomogeneous and anisotropic material coating that bends the incident wave around the object encompassed inside the coating, such that the object cannot be observed from the outside [7]. Inspired by the original cloaking design [2,8], many variations of cloaks have been proposed using modified coordinate transformations, including cloaks with different shapes and simplified material parameters [9–12], directional cloaks [13,14], conformal mapping-based isotropic cloaks [15], complementary media-enabled cloaks at a distance away from the object [16], localized cloaks that hide only a part of the object [17], carpet cloaks that give much broader operational bandwidth [18–23], macroscopic cloaks that hide electrically large objects [24–26], broadband cloaks for non-Euclidean space [27], all-dielectric cloaks [28–30] and so on. The more general illusion device, however, transforms the scattering perception of an object into that of another pre-defined target object [31–33]. It should be noted that cloaking is a special case of illusion optics, where the target object is free space. Apart from cloaking and illusion devices, other transformation optics-based designs for functionalities that were unable or difficult to achieve in the past have also been reported. These include polarization rotators [34–36], field concentrators [37–39], sharp waveguide bendings [40–42], wave collimators [42–45], beam splitters [35,46], cylindrical superlenses [47], wavefront shapers [48], controlled Cerenkov radiation [49], flattened and thin Luneburg lenses [50–52], superscatterers [53,54], field shifters [46,55], flat far-field focusing lenses [42,56], flat reflectors [57,58], optical ‘Janus’ devices [59], broadband enhanced optical transmission [60,61], multi-beam antenna lenses [62–65], beam expanders [66,67], low-profile antenna arrays [68–70], omnidirectional retroreflectors [71], omnidirectional absorbers [72,73] and so on.

Although transformation optics provides the general path to the design of a medium with exotic electromagnetic effects, the experimental implementation is a non-trivial challenge. Fortunately, with the development of metamaterial technology, previously inaccessible electric and magnetic material properties can be achieved by artificially structured building blocks [74–76]. By tailoring the resonant or non-resonant electric and magnetic responses of these subwavelength building blocks, which are typically arranged in a periodic fashion, negative [77,78], near-zero [79–81] or high [82,83] permittivity and permeability values can be realized along with a controllable degree of anisotropy [84,85]. The required inhomogeneous and/or anisotropic material properties of the transformation optics designs can thus be realized using specially designed metamaterials. However, because of the fact that the material parameters of most of the above-mentioned transformation optics designs have extreme values, which make their metamaterial realizations highly sensitive to frequency shifts and absorption losses, the majority of these designs are either limited to mathematical constructions and numerical validations or have only been experimentally demonstrated within a narrow frequency range. Several efforts have been carried out in recent years to mitigate these shortcomings by sacrificing certain types of functionality. One approach is to employ quasi-conformal mappings [18], which eliminates the anisotropy and non-unity permeability of the transformed medium because the magnetic response at optical wavelengths is very rare and weak even in optical metamaterials. The drawback of such methods is that the transformed medium would suffer from increased reflection on the interface between itself and free space. This, however, can be mitigated by immersing the device in a background medium with a specific dielectric constant higher than that of free space. Hence, under such circumstances, non-resonant metamaterials can be used to realize the design, resulting in a broad operational bandwidth [19,50]. In order to further reduce loss, dielectric blocks with drilled holes were employed for both microwave [51] and optical [20,21,23] transformation optics devices. The other approach, instead of reducing anisotropy, is to reduce the inhomogeneity of the transformed medium [24,25,63]. This simplification can be achieved by employing linear functions in the coordinate transformation, which simply involves expanding, compressing and rotating the coordinates. As a result, homogeneous anisotropic metamaterials or even natural crystals can be used to realize broadband transformation optics devices, at the expense of a limited device field-of-view.

In this paper, we first review the theory of transformation optics through an example of a far-field focusing lens. We then show that, in §3, more comprehensive control over both the phase and amplitude of the waves can be accomplished by a complex coordinate transformation. In §4, a linear transformation is presented, which achieves a similar directive emission functionality, but with simplified material parameters, making possible the broadband operation of the device. Experimental results are also shown to validate the concept. In §5, a tunable simplified transformation optics far-field focusing lens is introduced and implemented in the microwave regime, which achieves active beam steering. Transformation optics designs that control the flow of surface waves at optical wavelengths are then illustrated in §6 and §7, using either gradient-index materials or graphene sheets. Finally, the conclusion is drawn in §8.

2. Theory of transformation optics

As one of the properties of Maxwell's equations, which govern all electromagnetic phenomena, their form invariance under coordinate transformations provides the theoretical foundation of transformation optics. Assuming no electric and magnetic sources, in a Cartesian coordinate system, the electric field and magnetic field , with a time dependence of , satisfy Maxwell's equations as

2.1

where and are the permittivity and permeability tensors, respectively. A new coordinate system (x′,y′,z′) can be obtained by applying a transformation to the original coordinate system (x,y,z) with x′=x′(x,y,z), y′=y′(x,y,z) and z′=z′(x,y,z). It can be rigorously proven that, after such a coordinate transformation, Maxwell's equations maintain the same format as in the new coordinate system, which can be expressed as [1–3]

2.2

In this transformed coordinate system, the new permittivity and permeability tensors and are related to the original ones according to

2.3

where Λ is the Jacobian matrix of the transformation from (x,y,z) to (x′,y′,z′) defined as

2.4

This Jacobian matrix is a measure of the geometrical variation, such as expansion, compression and rotation, between the original space and the transformed space locally at every point. The original and transformed spaces are sometimes also referred to as the virtual space and physical space, respectively. The electric and magnetic fields in the transformed space can also be related to the values in the original space as

2.5

Such properties hold not only for Cartesian coordinate systems, but also for other curvilinear coordinate systems, including cylindrical and spherical coordinate systems. Equation (2.1)–(2.5) set down the mathematical foundation of transformation optics, illustrating the relation material properties and the geometrical properties of the two spaces [86]. It ensures that the physical characteristics of electromagnetic waves are preserved, even though the operating coordinate system is changed. To design a transformation optics device, one would follow a three-step procedure. First, one defines the desired wave propagation characteristics in the virtual space (x,y,z) with the constitutive parameter tensors denoted as and . Next, the designer finds a physical space (x′,y′,z′) with the constitutive parameter tensors denoted as and , which are unknown. In order to maintain the same wave propagation characteristics as that of the virtual space, the material parameter tensors of the physical space can be calculated in the last step using equations (2.3) and (2.4).

As an illustrative example, a flat far-field focusing lens that produces a collimated wave from a single electromagnetic source is presented here, which was previously proposed in [42]. Without loss of generality, considering the transverse electric polarization where the electric field is always oriented along the z-direction, a line source along the z-axis will produce a cylindrical wave in the x–y plane. As shown in figure 1a, the virtual space has equi-amplitude equi-phase lines, which are a group of concentric arcs with equal distances in between them, indicating the external diverging cylindrical-wave characteristics. In order to convert this cylindrical wave into a collimated plane wave, the equi-amplitude equi-phase lines have to be gradually transformed from circles to straight lines, as shown in the physical space in figure 1b. As a result, the coordinate transformation maps a section of a circle with a radius of a into a rectangle with a width of 2w and a thickness of t at a distance of g away from the x-axis, which can be expressed mathematically as

2.6

where a²=g²+w². By using equations (2.3) and (2.4), the permittivity and permeability tensors in the physical space are found to be

2.7

To validate the design, full-wave numerical simulations using COMSOL, which is based on the finite-element method, were carried out with the dimensions of the lens as w=4λ₀, g=0.5λ₀ and t=λ₀. A snapshot of the z-directed electric field is shown in figure 1c. It can be observed that the flat transformation optics lens indeed transforms the cylindrical wave emanating from the line source into a plane-wave-like radiated beam propagating in the+y direction. In the region behind the source, a cylindrical-wave-like pattern can be seen. The small ripples occurring in the collimated wave imply that slight reflection is caused by the truncation of the physical space. It should be noted that, in order to reduce the reflection of the flat lens, modifications can be made to the geometry of the physical space to improve its impedance matching to free space [56]. Such an example shows that one can indeed use transformation optics to design an electromagnetic device for manipulating the propagation of waves and thus achieving, in this case, wavefront control. It also proves that the design procedure outlined above is an efficient way for deriving the correct and complicated spatial distribution and anisotropy of the required material properties to fulfil the desired functionality.

Figure 1.

(a) The virtual space comprised a semicircular region. (b) The physical space comprised a thin rectangular slab. (c) Snapshots of the simulated total electric field distribution due to a z-directed electric line source located at the origin for a specific complex coordinate transformation with g=0.5λ₀, w=4λ₀ and t=1λ₀. (Adapted from [42].) (Online version in colour.)

3. Flat lenses designed by complex coordinate transformation

Most of the previously reported transformation optics designs, including the invisibility cloaks, illusion optics devices and the far-field focusing lenses, only use transformations between coordinate systems having real-valued coordinates. Recently, a new transformation optics approach was proposed, which is based on complex coordinate transformations [87]. It allows for the control over not only the phase but also the amplitude of the electromagnetic fields by purposely introducing non-vanishing imaginary parts into the coordinates in the transformed space. The introduced imaginary parts of the coordinates are then transferred into non-zero imaginary parts of the resulting permittivity and permeability tensors in the transformed medium. Consequently, the imaginary components of the material parameters can be tailored to provide either gain or loss to the transformation optics devices for simultaneous amplitude and phase manipulations. This capability, which can be of particular usefulness in antenna applications, has not previously been possible with the conventional transformation optics approaches.

To demonstrate its advantage over the conventional coordinate transformation method using real-valued coordinates, in this section, we extend the work on the flat far-field focusing lens by incorporating a complex coordinate transformation into the design. It is well known that the near-field aperture distribution controls the far-field radiation pattern in terms of both the beam width and side lobe level (SLL), which are important parameters for antenna engineering [88]. The aim here is to use complex coordinate transformations to provide a means for tailoring the aperture field distribution on the top interface of the lens, thereby shaping the radiation pattern in the far-field. Using the material parameters in equation (2.7) as a starting point, a smooth scalar function was introduced which is expressed as

3.1

where α is a constant factor that controls the degree of tapering. The desired near-field can be expressed by multiplying equation (3.1) with the original electric field generated by the source, shown as

3.2

As the design goal was to create a Gaussian profile in the near-field across the aperture, which is along the x-direction, the complex coordinate can be adapted and rewritten as

3.3

As we are considering the transverse electric polarization, which indicates that the electric field only has a z-component, ρ_im can be further simplified to

3.4

Substituting equation (3.4) into equation (3.3) and applying the resulting expression to the set of equation (2.6) yields the complete complex coordinate transformation as

3.5

The material parameters in the flat lens region, i.e. the transformed space, can consequently be obtained. The final results are given by

3.6

The real and imaginary parts of the material parameters across the lens are plotted in figure 2d. It is worth noting that, when α=0, there is no loss introduced into the material and the complex coordinate transformation design reduces to the same design as was obtained via the conventional phase-only coordinate transformation. Moreover, the smooth scalar function M can be arbitrarily chosen, such as a higher order function of x′, to achieve different amplitude and phase tapering profiles.

Figure 2.

(a) A snapshot of the simulated total electric field distribution due to a z-directed electric line source located at the origin for a specific complex coordinate transformation with α=−2, g=0.5λ₀, w=4λ₀ and t=1λ₀. (b) Simulated far-field patterns for α=0, −2, −5 and −7 with the geometry shown in (a). (c) Simulated gain, maximum minor lobe level (MMLL), side lobe level (SLL) and end-fire lobe level (EFLL) variations as a function of the geometrical dimension a in the original space with α=−2. (d) Components of the material parameters (real and imaginary parts) for the far-zone focusing lens with α=−2. (Adapted from [89].) (Online version in colour.)

The complex coordinate transformation lens was simulated using COMSOL. The snapshot of the total electric field distribution due to a single z-directed line source located outside the lens is shown in figure 2a for α=−2. When comparing with the total electric field distributions of α=0 (see figure 1c), the electric field for the α=−2 case is flatter and hence results in a narrower far-zone beam width (see figure 2b). Several full-wave simulations have been conducted for different values of α with α=−5 and −7. We define the SLL as the maximum value of the first side lobe relative to the peak of the main beam, and the end-fire lobe level (EFLL) as the maximum of the end-fire radiation relative to the peak of the main beam. As shown in figure 2b, as α decreases, the SLL drops while the gain is reduced as the half-power beamwidth (HPBW) gets wider. The gain values for α=0,−2,−5 and −7 are 12.81,11.59,9.93 and 8.99 dBi, whereas the SLL values are −11.32,−14.02,−19.42 and −24.58 dB, respectively. The HPBW values are 5.8°,6.2°,7.1° and 7.3°, respectively. As α decreases, the aperture field amplitude experiences an increased tapering while the phase becomes smoother and flatter [89]. In addition, as can be observed from figure 2b, the loss introduced by applying the complex coordinate transformation can reduce the ripple effect caused by reflection from the lens edges.

The impact of the lens geometrical dimensions on performance was also studied. It was found that increasing the lens width leads to an increase in gain, a decrease in HPBW, a decrease in EFLL and a decrease in the SLL. However, changing the thickness of the lens has almost no effect on the gain, HPBW, SLL or EFLL. An optimization on parameter a with a fixed w, which controls the boundary of the virtual space, was conducted in order to obtain maximum gain and minimum SLL. As shown in figure 2c, the optimal result of a can be obtained by maximizing the quantity {Gain(a)+| MMLL(a)|}, which yields an optimal value of a=3.8385λ₀. This results in a lens with a gain of 12.02 dBi and an MMLL of −13.56 dB. The optimized design was further compared with a conventional linear array (not shown here). It was found that the optimized lens has almost the same gain as the linear array with slightly higher SLL. The EFLL of this lens is about 10 dB larger than that of the linear array. Compared with a uniform array of isotropic sources, the lens only requires one single source located outside of the lens. The conventional linear array has a strong main beam in both the forward and backward directions, whereas the lens produces unidirectional radiation, which is an advantage in certain applications.

4. Linear transformation-enabled broadband multi-beam lenses

Even though the far-field focusing lens design using the conventional transformation optics or complex coordinate transformation approach numerically achieves the goal, implementing it experimentally is a great challenge for this particular design. It can be observed that the required material parameters are quite complicated, involving inhomogeneous and anisotropic permittivity and permeability tensors with extreme values. To simplify the material parameters, alternative coordinate transformations can be used. As figure 3a shows, considering a two-dimensional directive emission transformation, where the fields are restricted to be invariant along the z-direction, the virtual space is an air-filled, fan-shaped region (OMN) with a central angle of θ₁. It is mapped to an isosceles triangle with a vertex angle of θ₂ in the physical space (OXY). Note that the value of θ₁ is much smaller than the value of θ₂ to correspond to high directivity in the virtual space. As the equi-amplitude equi-phase lines of the virtual space are a set of parallel arcs while those of the physical space are a set of parallel straight lines, a direct mapping from the fan-shaped region to the triangular region will result in inhomogeneous and anisotropic material parameters. To reduce the spatial dependency of the transformed material parameters, a geometrical simplification can be applied to the virtual space geometry. As the value of θ₁ is small, the fan-shaped region can be approximated by an isosceles triangle with the same central angle. By applying this simplification, the coordinate transformation has linear functions for all coordinates, resulting in homogeneous and anisotropic transformed material parameters. The permittivity and permeability tensors thus have a near-zero value in their x components, and a near-unity value for the y- and z-components, with zero off diagonal parameters. The low permittivity and permeability tensor parameters are in the direction of the directive radiated beams. By further applying the coordinate rotation transformation to the above directive emission transformation, a highly directive lens can be synthesized by surrounding the embedded isotropic source with several triangular segments to produce multiple directive beams simultaneously.

Figure 3.

(a) Two-dimensional directive emission coordinate transformation. (b) Snapshots of the z-directed near- and far-zone electric field determined via a two-dimensional COMSOL simulation of the transformation optics lens at 3 GHz. (c) Photograph of the fabricated lens. (d) Simulated and measured reflection coefficient of the monopole with the transformation optics metamaterial lens. The inset shows the electric field distribution on the ground plane with the lens present at 5.1 GHz. Simulated and measured (e) H-plane and (f) E-plane realized gain patterns of the monopole antenna with and without the transformation optics metamaterial lens at 5.1 GHz. (Adapted from [90].) (Online version in colour.)

The concept was validated by carrying out full-wave simulations using COMSOL, where, for simplicity, only TE polarization with a z-directed electric field was considered in the simulations. As figure 3b presents, the lens has four collimated beams uniformly distributed in the x−y plane pointing at ϕ={0°,90°,180°,270°}. Each lens segment has a low-value parameter with a magnitude of 0.01 in the direction perpendicular to its outer interface. From the electric-field distribution, it is observed that the waves radiated from the central isotropic source are well collimated, even in close proximity to the source. The lens produces four directive beams at the desired angles. Further studies showed that the lens performance is not sensitive to small material parameter variations, which makes the lens robust to fabrication tolerance. More importantly, as most metamaterial realizations of effective media are less dispersive in the low index band, which locates on the resonance tail [90–92], the insensitivity of material parameters makes this type of transformation optics lens suitable for broadband applications. The physical mechanism of such directive emission can be better understood by studying the dispersion relation of the lens medium. It should be noted that the anisotropy forces the waves to propagate close to the x-direction with a very small y-component in its wavevector. The behaviour is confirmed in figure 3b, where the waves are collimated not only in the far-field, but also in the near-field region inside each lens segment, even in close proximity to the source.

To verify the proposed multi-beam directive emission coordinate transformation experimentally, a quad-beam metamaterial lens was designed that reshapes the radiation pattern of a quarter-wavelength monopole operating at around 4–5 GHz. The monopole alone radiates omnidirectionally in the H-plane with the electric field nearly perpendicular to the ground plane, which is 14×14 cm² in size, in its H-plane. Hence, the lens only needs to work for the transverse electric polarization. Following the configuration of the two-dimensional quad-beam lens example in figure 3b, the monopole is surrounded with four triangular anisotropic metamaterial lens segments. The segments in the ±x and ±y directions have a low value of effective μ_rx and μ_ry, respectively. Broadside coupled capacitor-loaded ring resonators (CLRRs) made of copper [90] are used as the building blocks to produce the anisotropic magnetic resonant property, which gives a low permeability tensor parameter perpendicular to the plane of the CLRR. The openings of the CLRRs are oriented in opposite directions to eliminate any bi-anisotropy that might be caused by the structural asymmetry [93].

The metamaterial lens was fabricated and assembled. As figure 3c shows, it consists of nine concentric anisotropic CLRR layers. The square-shaped layers are each five unit cells tall and decrease in edge length by two unit cells for each successive layer starting from the outside of the lens and moving inward. The simulated and measured reflection coefficient (S₁₁) is shown in figure 3d, exhibiting a good agreement between simulations and measurements in terms of not only the −10 dB bandwidth but also the resonance positions. The measured reflection coefficient is smaller than −10 dB from 4.22 to 5.60 GHz with three resonances located at approximately 4.35, 4.7 and 5.45 GHz, exhibiting strong correspondence with the simulated results. Even at low frequencies, good agreement is observed in the minor resonances around 3.7 GHz and 3.95 GHz, which can be attributed to the near-field coupling to the lens. The realized gain patterns of the monopole antenna with and without the lens were characterized in an anechoic chamber with an automated antenna movement platform. The patterns were measured in a frequency range from 4 to 5.5 GHz, while, for succinctness, only the results at 5.1 GHz are presented (see figure 3e,f). Strong agreement can be seen between the simulated and measured results. The discrepancies are primarily due to minor inaccuracies in fabrication, non-ideal effects of the test set-up and noise in measurement. For the case without the metamaterial lens, the monopole has a nearly omnidirectional radiation pattern in this band. The maximum measured realized gain values in the H-plane vary between −1.6 and −0.3 dB. With the transformation optics lens present, four high gain beams are located at 0°,90°,180° and 270° with a measured realized gain varying between the range from 4.3 to 5.9 dB, yielding about 5.9–6.2 dB of realized gain improvement. The measured average HPBWs of the four directive beams are approximately 31°∼37° from 4.25 to 5.3 GHz. This indicates that the transformation optics lens has an operational bandwidth of over 20%, which is much broader than many of the previously reported transformation optics devices. At higher frequencies, the longitudinal effective permeability tensor parameter approaches unity, leading to an almost omnidirectional radiation pattern, even though the impedance is still well matched because of the electric response of the CLRRs and their near-field coupling with the monopole. The measured E-plane patterns confirm the beam bending effect in the θ-direction. Without the lens, the monopole alone has a measured beam maximum moving from 45° to 40° off horizon as the frequency increases. With the lens present, the beam maximum is maintained at approximately 8–12° from the horizon. In all, the experiment verifies the three-dimensional collimating effect of the metamaterial transformation optics lens in reshaping the radiation of the embedded monopole antenna into multiple highly directive radiated beams. The near-field distribution on the ground plane is also examined. As shown in the inset of figure 3d, a near-square pattern can be observed in the electric field distribution, indicating that the discrete metamaterial implementation of the lens indeed behaves as an effective homogeneous anisotropic medium with certain permeability tensor parameters having a low, near-zero value. Further simulation studies reveal that four directive beams can still be obtained even when the ground plane is truncated from 14×14 cm² to 11×11 cm², implying that the ground plane size has very little effect on the lens performance [90]. When the lens height is increased, higher gain can be obtained in the H-plane and the beam maximum will be shifted closer to the horizon.

5. Tunable conformal mapping lens for active beam steering

When combined with tunable or actively reconfigurable metamaterials [94], transformation optics-based design strategies can allow substantial improvements to device behaviour and performance. At the very least, the use of a frequency-tunable metamaterial to construct a transformation optics device can be used to increase the typically narrow operational bandwidth over the original, static design. Such examples might be demonstrated with uniform tuning of all unit cells throughout a lens to shift the channel frequency. A more interesting approach involves independent spatial control of the metamaterial properties throughout different regions of the lens which can be used to completely change the operation of the device. Depending on the metamaterial structure and tuning methodology, possible scenarios might include variation in index, anisotropy and material dispersion. Allowing the electrical properties of each metamaterial unit cell or subset of unit cells within a slab of metamaterial to be set independently effectively creates a slab with morphable and dynamic electrical geometry. With enough control over the effective material properties produced by each unit cell, such a slab would allow the implementation of a large number of possible transformation optics devices. Indeed, simplifying the allowed material parameter ranges, while reducing the flexibility of the resulting transformation optics devices, can greatly simplify the design, as well as the construction of a prototype.

Although there are many possible implementations that would be interesting applications of a spatially reconfigurable transformation optics device, the example considered here is intended to extend the capability of a simple near-zero-index metamaterial lens antenna system [62]. Different from the lens presented in the previous section, where a linear transformation was used to simplify the material parameters, a conformal mapping can also be employed to reduce the inhomogeneity of the permittivity and permeability tensors of the transformed medium. It has been shown that conformal mappings make possible the realization of an anisotropic zero-index metamaterial lens for directive emission. Different from the lens obtained using linear transformations, the conformal mapping-enabled anisotropic, zero-index metamaterial lens has its low permittivity and permeability parallel to the polarization of the electromagnetic source. Considering transverse electric polarization, where a line source is located along the z-axis, in order to create the directive emission lens requires that the z-component of the permittivity tensor possesses a near-zero value. The geometrical shape of the lens and the location of the source determine the direction, as well as the number of the collimated radiated beams [62].

The integrated active antenna system is constructed by placing a vertically polarized magnetic dipole source within the centre of a slab of a homogeneous anisotropic zero-index material. When excited by the central feed, the structure's in-plane radiation pattern is determined by the geometry of the slab—a single high-directivity beam is radiated from each face of a polygonal slab, with larger faces producing more powerful and more collimated beams. Two examples can be seen in figure 4a,b. Combining this concept with the idea of a spatially reconfigurable metamaterial slab where each unit cell switches between near-free-space and near-zero-index enables an antenna system that can produce multiple beams in arbitrary directions by changing the effective shape of the anisotropic zero-index metamaterial lens, as in figure 4c. The reconfigurable metamaterial allows tuning of the electrical shape of the slab, and thus changing of the radiation pattern of the antenna in real time, without the use of discrete phase shifters as in a phased array or mechanical actuators as in a scanned reflector antenna. The planar metamaterial lens is truncated to a finite thickness to form a cylindrical slab metamaterial lens, illustrated in figure 4d with a representative radiation pattern in figure 4e.

Figure 4.

(a–c) Planar near-zero-index lenses produce radiation patterns controlled simply by the shape of the anisotropic zero-index material slab, allowing (a) quad-beam, (b) three-beam and (c) arbitrary beam scanning enabled by a switchable metamaterial. (d) Extending the circular lens to three dimensions through the addition of metallic plates enables planar beam forming, with a representative radiation pattern shown in (e). (f) Conceptual spatially reconfigurable unit cell with switches and a shift register control circuit in the centre of a group of six SRR elements. (g) These hexagonal unit cells can be tiled in a space-filling curve to form the approximately cylindrical active metamaterial. (Online version in colour.)

The limiting factor for any reconfigurable transformation optics design is the metamaterial system used for its construction. The selected example of a metamaterial slab whose unit cells are independently tuned from near-free-space to near-zero-index is a good candidate for an initial implementation, as each unit cell requires only a single bit of control—on or off. Complex tuning that controls the magnitude of one or more components of the permittivity or permeability tensors requires sophisticated control circuitry beyond that of the simple switch which suffices for the reconfigurable anisotropic zero-index metamaterial slab.

A potential implementation for the reconfigurable lens antenna makes use of split ring resonators (SRRs) tuned with varactor diodes, which act as voltage-controlled variable capacitors. The SRR is a planar resonator that interacts strongly with the normally oriented magnetic field to form an artificial magnetic material [95] whose effective magnetic permeability is controlled by the geometric and electrical properties of the SRR array. Variants of this unit cell are commonly used to create negative-, zero- and high-index metamaterial devices. Placing a lumped capacitor in the split allows the resonance frequency, and thus the frequency of the zero-permeability condition, to be controlled [96]. With a varactor, the reverse bias voltage and thus the RF capacitance across the element are used to control the resonant frequency of the SRR elements in real time [97,98].

Assembling an array of switchable unit cells allows the properties of the slab as a whole to be changed by setting the bias voltage from the edge of the lens, but additional circuitry is required to allow independent spatial control of the resonators. The varactors require a power supply to set the bias voltage, and the state of each varactor should be independently controlled in order to allow completely arbitrary spatial tuning within the slab. A distributed shift register is an efficient method, in terms of the fabrication expense and the number of wires and traces required for implementation, for independent addressing of each resonator. Each resonator is associated with a single bit output in a series of chained, discrete shift registers. Changing the bit pattern provided at the input will set the state of each resonator, to be near-free-space or near-zero-index. A conceptual drawing of the individual unit cell and their interconnection into a spiral array is shown in figure 4f,g.

Depending on the frequency range of interest, the reconfigurable metamaterial implementation as described can be fabricated and assembled by standard printed-circuit board techniques and with off-the-shelf varactors and other components. In order to achieve significant directivities, the lens should be several wavelengths in diameter at the operational frequency, which requires a large number of unit cells to create such a large metamaterial panel. The final antenna consists of multiple layers of the planar resonators stacked vertically to form a volumetric metamaterial array, where the top and bottom of the lens are covered by metal plates and the radiation pattern is controlled in the plane of the lens. After placing a magnetic dipole feed at the centre of the lens, the radiation pattern of the resulting antenna is set by changing the bit pattern of the control signal at the input of the lens to generate one, two or more beams.

This beam-steering antenna system is just one example of the new devices that are possible when leveraging and combining the active metamaterial and transformation optics design paradigms. With slight extensions to the metamaterial design described above, each unit cell could employ two or more bits of the shift register to enable magnitude tuning of the effective material response, rather than only on–off control. In this way, the metamaterial slab may be tuned to produce a number of different transformation optics devices with the same physical structure. Active research is ongoing to extend and enhance the behaviour of metamaterials in general and reconfigurable metamaterials in particular throughout the RF spectrum.

6. Photonic integrated circuits designed by quasi-conformal mappings

In the optical regime, the range of available metamaterial implementations becomes limited by more stringent fabrication constraints compared with radio frequency devices, and anisotropic materials become increasingly difficult to locate or construct. The reliance of many transformation optics designs on anisotropic metamaterials for implementation substantially limits the range of possible transformation optics devices constructed for optical applications. Transformation optics devices that have been constructed for optical wavelengths can typically be either built as a significant approximation to the analytical index distribution or designed using special coordinate transformations that yield simple material parameters.

Conformal mappings are a class of two-dimensional coordinate transformations in the complex plane that satisfy a number of constraints, such as mesh orthogonality and preservation of angles between the source and effective coordinate systems [99]. The constraints of a conformal mapping reduce the design space of possible transformations and thus wave and field behaviours, but in turn allow implementation of the resulting transformation optics device with an isotropic all-dielectric gradient-index profile. Devices with all-dielectric gradient-index profiles are much simpler to fabricate in the optical, infrared and terahertz wavelengths than the highly anisotropic profiles of an ordinary transformation optics design. Several synthesis approaches exist for the creation of these gradient-index devices, including the synthesis of an effective dielectric coefficient through metamaterial approaches, controlled mixing of two materials and drilling nanoscale air holes through a substrate in appropriate sizes and densities to realize an effective index gradient [20,21].

Although conformal mappings require analytical derivations for an exact solution, they may be closely approximated numerically by the so-called quasi-conformal transformation [100] that relaxes several constraints while maintaining the beneficial properties of conformal mappings for transformation optics. Quasi-conformal mappings can be easily adapted to optimization-based design algorithms and numerical evaluation of device behaviour. As in any transformation optics problem, a quasi-conformal mapping design begins by selecting the source and destination coordinate systems for which the transformation will be defined. For a quasi-conformal mapping, the coordinate systems and the associated transformation are entirely generated by selecting the boundary contours of two regions in the complex plane to be the source and destination domains. For example, by selecting the source to be a circle and the destination to be a rectangle, then the mapping will generate an index profile that redirects energy from a point source located at the origin into collimated beams from each of the faces of the rectangle. The shapes of the two domains and the position of the source (internal, external) can be selected to create a wide range of possible transformation devices. Constraints on the allowable manufacturing parameters (index ranges, materials, spatial and index tolerances) can be included in the design and optimization process to ensure that the resulting specifications are viable.

A typical optimization procedure for a quasi-conformal mapping lens would involve selection of the two domain contours. When designing a flat focusing lens, for example, the optimization might use a convex source domain and rectangular destination domain. By selecting the radius of curvature of both faces of the source domain, as well as the thickness and any aspherical surface profile perturbations, the lens properties are varied until the effective response of the lens, as computed by a full-wave solver or ray tracer, is deemed acceptable. Optimization may not be required for all designs, but use of a stochastic optimizer is vital when fine-tuning for high-performance optical systems, for example. Using such a technique, an on-chip beam collimator was designed based on the mapping from a circular virtual space to a square physical space, as illustrated in figure 5a. The four additional protrusions on the corners are chosen in order to facilitate orthogonal grid generation through quasi-conformal mappings. The resulting index profile exhibits fourfold symmetry and has high index at each collimating surface (figure 5b). The four faces of the physical space are connected to silicon waveguides. Figure 5c shows the light intensity distribution excited by a point source at the centre of the device, demonstrating efficient coupling from the light source to the four waveguides at 1.5 μm.

Figure 5.

(a) Coordinate mapping from the virtual space (top) to the physical space (bottom). (b) A quad-beam collimator connects four silicon waveguides. The inset shows the index profile. (c) Simulated light intensity distribution excited by a point source at the centre of the component at 1.5 μm. (d) Illustration of optical integrated circuit components constructed using quasi-conformal transformation optics. (Adapted from [100].) (Online version in colour.)

All-optical computing relies on optical fibres for controlling the distribution of light, but is limited in capability by the requirement to convert optical signals to electrical signals for processing and calculations. Adding optical processing components and constructing entire optical integrated circuits with the same bulk fabrication techniques used for the development of electrical circuits offers many advantages over conventional optical fibre implementations. Among other applications, the development of planar, all-dielectric optical components using quasi-conformal mapping that may be combined to form complex optical integrated circuits is an interesting and useful goal. The quasi-conformal mapping approach used to design the on-chip beam collimator can also be extended to synthesize other types of planar photonic components used in the integrated photonic circuit shown in figure 5d, including light collimators for fluorescing optical quantum dots, couplers to convert between larger and smaller waveguides, beam bends, beam splitters and a waveguide crossing circuit that allows two optical waveguides to cross signals without crosstalk (more details are available in [100]). Although these devices may also be designed using unconstrained transformation optics [6], the all-dielectric gradient-index profiles of the quasi-conformal mapping design approach offers a substantial benefit for practical applications. These represent some but not all of the possible ways that quasi-conformal mapping design principles may be used to produce and develop optical integrated circuits.

7. Surface wave manipulation using graphene

Apart from the dielectric waveguide, which serves as the basis for the photonic integrated circuits discussed in the previous section, at optical wavelengths, noble metals, such as gold and silver, support the propagation of surface waves, which are referred to as surface plasmon polaritons (SPPs) [101,102]. These materials suffer from significant losses at optical wavelengths that degrade the plasmon resonance and reduce the propagation lengths of the SPPs [103]. Furthermore, it is difficult to control the material parameters of such mediums, limiting their potential for realizing gradient-index material distributions for transformation optics-inspired devices. In this section, we illustrate how transformation optics designs can be achieved based on graphene sheets to control the propagation of SPPs with low loss.

Recently, graphene, an allotrope of graphite, has been proposed as a suitable host for novel planar gradient-index devices, such as Luneburg lenses [103] and convex lenses [104]. Previous authors have experimentally demonstrated the possibility for incorporating non-uniform patterns with the graphene sheet's electronic transport properties [105]. This is achieved in a number of ways, including manipulating the graphene's substrate thickness, substrate permittivity or bias voltage. Graphene is a particularly attractive host medium in a variety of optical systems, including integrated photonic components [100], gradient-index lenses and various optical metamaterials, owing to its single-atom thickness, limited footprint and highly tunable electronic properties. These enable the design of miniaturized, high-functionality devices that can be integrated on a single graphene sheet.

The utility of graphene as a host for gradient-index devices largely relies on the imaginary component of its complex conductivity σ_g as

7.1

which depends on the angular frequency ω, particle scattering rate Γ, temperature T and chemical potential μ_c. This dependence is embodied by the intra-band and inter-band conductivity, which are expressed as [106]

7.2

and

7.3

where k_B is Boltzmann's constant and is Planck's constant. The inter-band conductivity term is an approximate expression that relies on .

The correspondence between the conductivity of the graphene host and the SPP is dictated by the dispersion relation. The resulting material tensors from the transformation optics calculation are directly related to the complex conductivity of graphene. From the dispersion relation of a transverse magnetic SPP

7.4

we can approximate [103]

7.5

where β and k₀ are, respectively, the wavenumbers of the guided mode and of free space, while η₀ is the intrinsic impedance of free space and n_SPP is the effective index of refraction of the SPP. From equation (7.5), we observe that when σ_g,i>0 the graphene host supports transverse magnetic SPP waves, but when σ_g,i<0 the host does not support the modes. This indicates that, with quasi-conformal mapping design techniques, the resulting dielectric-only distributions will produce corresponding conductivity values that support transverse magnetic SPP waves. An additional constraint for designing transformation optics-inspired devices on graphene is that the background permittivity of the system when performing the transformation must be chosen such that the resulting conductivity corresponds to achievable values.

The imaginary component of the complex conductivity is of particular interest, as it relies on the graphene sheet's chemical potential, which is governed by the carrier density [107]

7.6

and

7.7

where ν_f is the electron velocity. This expression indicates that manipulating the substrate permittivity ϵ_d, the gate voltage V_b or the thickness of the substrate d allows designers to control the sheet's conductivity, enabling the development of gradient-index profiles. These design equations provide the essential link for relating the necessary electronic transport properties of graphene to the material tensors resulting from quasi-conformal mapping techniques. The procedure outlined above offers the foundation necessary for developing gradient-index devices on graphene hosts.

The design equations and procedures discussed above can be used to realize novel optical devices on a graphene host. Here we demonstrate the utility of the quasi-conformal transformation optics (QCTO) technique to design a planar quad-beam collimator based on a graphene host. The quad-beam collimator efficiently couples wave fronts from a two-dimensional isotropic source at its origin into highly collimated wave fronts at each of its interfaces.

To realize the flattened interface, quasi-conformal mapping techniques are used. These techniques produce nearly-isotropic and dielectric-only materials. This indicates that the refractive index of the SPP in equation (7.5) can be represented solely by the contribution of the permittivity, . The transformation distorts the four arcs of the circle into straight edges of a square. The resulting effective permittivity distribution is shown in figure 6a. The background permittivity when extracting the material tensors is appropriately selected to provide a desired range of graphene conductivity values. From the effective permittivity (or impedance), the corresponding imaginary conductivity values can be calculated, as shown in figure 6b, using the fact that n_SPP=ϵ_eq. The chemical potential is then computed assuming that Γ=0.43 meV, T=300 K and f=30 THz. Then, from the chemical potential, designers have the choice of numerically solving for the required substrate permittivity, thickness or bias voltage using equations (7.6) and (7.7) to achieve the desired GRIN profile. In this case, we solve for the required substrate permittivity distribution, shown in figure 6c, assuming d=50 nm and V_b=25 V. This range of values can be effectively reduced by clipping the extreme regions at the corners without significant performance degradation. Here, the graphene layer can be designed to be supported on SiO₂ (ϵ_r=3.9).

Figure 6.

(a) Effective permittivity distribution obtained from quasi-conformal mapping techniques. (b) Corresponding graphene imaginary conductivity using equation (7.5). (c) Required graphene substrate permittivity to achieve desired gradient-index distribution. (d) Normalized electric field distribution of a quad-beam collimator device. (Online version in colour.)

The performance of this device was then evaluated through full-wave simulation. COMSOL Multiphysics was employed to perform the full-wave simulations and demonstrate the performance of the collimator. Since the software package cannot implicitly model graphene electronic transport properties, the domain was populated with the effective permittivity distribution, as shown in figure 6a. This effective material distribution is representative of the electronic properties on the graphene sheet so that the simulation replicates the propagation of SPP waves. The system is excited with a 30 THz line source perpendicular to the plane. Each of the four interfaces is terminated with a waveguide to support the propagating mode. It is clear from figure 6d that the gradient profile efficiently collimates the circular wave front into a planar wave front. Furthermore, this functionality is accomplished within a few wavelengths of the epicentre. The interference in the system originates from reflections due to impedance mismatches at the boundaries of the device.

8. Conclusion

In conclusion, we have reviewed the recent advances in the field of transformation optics. Its theory was revisited through both the basic mathematical expressions and a concrete electromagnetic design example. Several advanced transformation optics design techniques were then presented including the complex coordinate transformation, linear coordinate transformation, conformal mapping electrically tunable transformed space geometry for microwave antenna systems and quasi-conformal mappings for optical applications. Numerical and experimental examples were illustrated which validate the concept as well as the specific transformation optics-inspired electromagnetic designs. It has been shown that these techniques evolved from the conventional transformation optics approach and provide, in different aspects, an improved device performance and enhanced functionality from the microwave regime to optical wavelengths. Table 1 compares the properties among different coordinate transformation methods in terms of the required material properties, operational bandwidth, the ease of realization in microwave, terahertz and optical ranges, and their associated advantages. For a specific task, one method may clearly emerge as being more appropriate than the others. A hybrid method that combines two different methods may even be desirable for some complicated problems under certain constrained conditions. Above all, by virtue of the powerful capability of transformation optics in manipulating electromagnetic waves, it will continue to evolve and inspire advanced integrated electromagnetic devices.

Table 1.

Comparison among different methods of coordinate transformation.

	material properties	bandwidth	realization	advantages	disadvantages
conventional coordinate transformation	inhomogeneous+anisotropic+passive	narrow	easy at microwave and terahertz, difficult at infrared/visible	exact solution without sacrifice of performance	extreme parameter requirements and absorption losses
quasi-conformal mapping	inhomogeneous+isotropic+passive	very wide	easy in all spectral ranges	low loss and only indices greater than unity required	interface impedance mismatch and slight material anisotropy
linear transformation	homogeneous+anisotropic+passive	wide	easy at microwave and terahertz, implementable at infrared/visible	robust to variation of material properties	limited device field-of-view
complex coordinate transformation	inhomogeneous+anisotropic+passive/active	narrow	easy at microwave, difficult at terahertz and infrared/visible	simultaneous amplitude and phase control	extreme parameter requirements and losses

Competing interests

The authors declare no competing financial interests.

Funding

This work was supported in part by NSF MRSEC under grant no. DMR-0820404.