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We describe the design of adaptive beam bends and beam splitters with arbitrary bend and split angles by use of finite embedded coordinate transformations. The devices do not exhibit reflection at the entrance or exit surfaces. It is shown that moderate and practically achievable values of the relative permittivity and permeability can be obtained for beam bends and splitters with both small and large bend radius. The devices are also discussed in the context of reconfigurable metamaterials, in which the bend and split angles can be dynamically tuned. The performance of adaptive beam bends and splitters is demonstrated in full wave simulations based on a finite-element method. Furthermore, the design of an adaptively adjustable transformation-optical beam expander/compressor is presented. It is observed that a pure transformation-optical design cannot result in a reflectionless beam expander/compressor.
©2008 Optical Society of America
1. Introduction
Transformation optics offers an interesting and in some ways unique approach for the design of complex electromagnetic materials with unconventional optical properties. The general method is based on form-invariant coordinate transformations of Maxwell’s equations [1, 2]. In order to design a transformation-optical element, a specific coordinate transformation is deployed that maps an original space with given electromagnetic material parameters – like the electric permittivity and magnetic permeability – and a given metric into an envisioned space. As a direct consequence of the coordinate transformation, the permittivity and permeability, as now described in terms of the new coordinates, have to be transformed accordingly. The pivotal step in the investigations of Pendry et al. was the alternative interpretation of such coordinate transformations [1]. Instead of considering a coordinate map as a pure transition to a new coordinate system – which would not change the underlying physics – the transformed quantities of the permittivity and permeability can be regarded as the “real” material properties of a new medium in the original, untransformed coordinate space. This alternative interpretation, also called the “material interpretation” [3], provides a powerful design method for optical elements with complex material parameters and unique functionality. The transformation-optical approach is especially compelling as a sophisticated design approach that fully exploits the freedom in the choice of the complex permittivity and permeability tensors in metamaterial structures. Without such a systematic methodology, the designer of metamaterial devices would be most likely overwhelmed and lost in an almost unlimited parameter space. Conversely, transformation optics would be meaningless if artificial materials like metamaterials with configurable electric and magnetic response were not available for the practical implementation and fabrication of the designed elements.
One of the most striking examples of an unconventional transformation-optical device was the design and experimental demonstration of an “invisibility cloak” at microwave frequencies [4]. The fascinating and physically interesting nature of electromagnetic cloaks defined the focus of subsequent research in the field as indicated by many publications concerned with the extension and analysis of cloaking structures [5–20]. However, transformation optics is not only a powerful tool for the design of electromagnetic devices. In many cases the same or slightly modified principles can be applied to the design of acoustic metamaterial cloaks [21–25] and, as most recently suggested, for the cloaking of matter waves [26].
Although in the majority of publications transformation optics is solely used for the design of invisibility cloaks, the approach is a quite general technique for the development of innovative optical elements. Based on the mathematical formalisms provided by differential geometry, the transformation-optical methodology relates to general relativity [27] and quantum optics [28, 29] and can be used e. g. for the design of field concentrators [19, 20], emission control optics for electromagnetic sources [30], field rotators [31], the design of magnifying lenses with subwavelength resolution [32–34], subwavelength waveguides [35] and improved perfectly matched layers [36] for numerical studies.
Recently, we reported an expansion of the transformation-optical approach from continuous coordinate transformations to finite embedded transformations. As discussed in [37], finite embedded transformations allow the field manipulations performed by a transformation-optical medium to be transferred to the wave that exits the medium. This is in contradistinction to most transformation-optical designs presented, which are inherently “invisible”. Although the finite embedded approach does not inherently result in reflectionless devices in contrast to optical elements designed by continuous coordinate transformations, reflections at the boundaries of the transformation-optical medium can be completely avoided for a certain class of transformation optics. It was found by heuristical means that transformation-optical elements designed by finite embedded coordinate transformation can be reflectionless if the metric of the transformation-optical medium in the direction normal and parallel to the interface between the transformation-optical medium and the surrounding medium is continuous at the boundary.
In the following we will describe further interesting transformation-optical devices designed by finite embedded coordinate transformations. In section 2, we will explain the concept of transformation-optical beam bends and beam splitters with arbitrary bend and split angles. The bend angle hereby is defined as the angle between the direction of propagation of the incoming wave and the exiting wave from the transformation-optical medium. The split angle is accordingly the angle between the propagation directions of the split waves. The coordinate transformation for beam bends affects both dimensions in the plane of wave propagation resulting in the beam bend to tilt the wave fronts of the incoming beam. We will show that beam bends with almost arbitrary radius of curvature can be realized using metmaterials with moderate values of the permittivity and the permeability. We argue that reconfigurable metamaterials hold the key for reflectionless beam bends and beam splitters with tunable bend and split angles. This is demonstrated in full wave simulations which reveal the adaptive behavior of such transformation-optical devices.
In section 3 we provide the design of a transformation-optical beam expander/compressor. It is shown, that such components are not reflectionless if only the pure transformation-optical approach is applied. This is important since it shows the necessity to combine transformation optics with other classical approaches in order to optimize the performance of transformation designed optical elements.
2. Transformation-optical beam bends
In the following transformations, we assume free space with isotropic relative permittivity and permeability tensors εij=μij=δij as original space, if not otherwise noted. For a given coordinate transformation x i′=A i′ i xi, the relative permittivity and permeability of the transformation-optical medium are calculated by the prescription
where A i′ i=∂x i′/∂xi indicates the Jacobi matrix of the transformation and det(A i′ i) its determinant.
A possible coordinate transformation of a beam bend with inner radius R 1 and outer radius R 2 can be described in cartesian coordinates (x,y,z) by
where x∈[R 1,R 2], y∈[0,R 2/k], α denotes the total angle by which the beam is bent in the transformation-optical medium, sgn is the signum function and k∈(0,∞) is a free scaling parameter. Under this convention, an electromagnetic wave is steered clockwise for α<0 and counterclockwise for α>0.
As an example, Fig. 1 illustrates the coordinate transformation of a (α=π /5)-bend for k=1 (Fig. 1(a)) and k=2 (Fig. 1(b)) as described by Eq. (3)–(5) with α>0. As can be clearly seen, a global consideration of the transformation reveals space discontinuities at the boundary of the beam bend with reference to the surrounding euclidian space. At this point, we strictly follow the approach of finite embedded coordinate transformations as proposed in [37]. By applying this methodology, the boundaries are not taken into account in the calculation of the material parameters of the transformation-optical medium. The transformation is carried out exclusively in the local region of the transformation-optical medium and the obtained material is then embedded into free space. As pointed out in [37], the approach of finite embedded transformations allows the design of reflectionless optical elements under proper conditions regarding the metrics of the transformation.
Fig. 1. Illustration of the embedded coordinate transformation for a π/5-bend with (a) k=1 and (b) k=2.
In order to calculate the effective permittivity and the permeability of the transformation-optical beam bend, it is convenient to express Eq. (3)–(5) in a cylindrical coordinate basis. As a first step, we transform the euclidian space from the canonical cartesian coordinate system {x,y,z} into a cylindrical system {r′,ϕ′,z′} by applying the general transformation
with the corresponding inverse transformation
where ϕ∈[0,2π). The general coordinate transformation for an α-beam bend with cylindrical symmetry and inner radius R 1 and outer radius R 2 can then be written as
with
The electromagnetic properties of the transformation-optical medium can now be readily obtained by calculation of the Jacobi matrices of Eq. (6)–(8) and (12)–(14), substitution into (1)-(2) and multiplication of the inverse matrix of the obtained permittivity and permeability tensors for euclidian space with the corresponding tensors for the transformation-optical medium (see a detailed discussion of this procedure in [19]).
Fig. 2. (a) Relative permittivity and permeability η=ε=μ in the radial (ηr), azimuthal (ηϕ) and in the z-direction (ηz) in dependence on the radial distance rnorm=r/R (R: arbitrary) from the origin of an α-bend for various values of γnorm. (b) Illustration of the contrivable physical dimensions of an α-bend with arbitrary α for various values of γnorm. The inner radius R 1 and the outer radius R 2 of the bends were chosen such that ε<10 and μ<10.
Following this recipe, one obtains for the permittivity and permeability tensors of the beam bend expressed in a cylindrical coordinate system
with γ=R 2/(kα). Note that r″ was substituted by r as the material properties are expressed in the “material interpretation” [19]. Since the same α-bend with inner radius R 1 and outer radius R 2 can be realized by an infinite number of possible choices of the factor k and the permittivity and permeability in (16) are only dependent on r and k, k can be used as a free parameter to keep the values of the material parameters within a convenient range.
Figure 2 displays the relative permittivity and permeability η=ε=μ of an α-bend for different values of γnorm=γ/R (R: arbitrary) with dependence on the normalized radius rnorm=r/R as calculated from Eq. (16). Hereby, the normalization radius R was arbitrarily chosen. For instance, one can assume that the beam bend extends from 0 to R in the radial direction and 0 to α in the azimuthal direction. The values of the relative permittivity and permeability in the radial and the z-direction diverge for rnorm→0 while the permittivity and permeability in the azimuthal ϕ-direction linearly depend on the distance rnorm from the origin of the beam bend. In the following we show how the free parameter γnorm can be used to achieve beam bends with different radius and width while maintaining the permittivity and permeability of the required metamaterial at moderate values. For γnorm=3, ηr and ηz possess moderate values ηr=ηz∈[2.5,10] while ηϕ∈[0.1,0.4] for rnorm∈[0.3,1]. For increasing γnorm, the values of ηr=ηz significantly decrease even for smaller rnorm and the values of ηϕ rapidly increase for increasing distance rnorm from the inner boundary of the bend. At γnorm=0.3, ηr=ηz∈[0.3,10] and ηϕ ∈[0,3] for rnorm∈[0.03,1]. As becomes obvious from Fig. 2, γnorm has to be chosen very small if a beam bend with a small inner radius R 1 and outer radius R 2 has to be realized while γnorm should be large if beam bends with large inner and outer radius need to be fabricated in order to achieve moderate electromagnetic material parameters. This is illustrated in Fig. 2(b) where the corresponding physical dimensions of an α-bend are plotted for various values of γnorm. The inner radius R 1 and the outer radius R 2 of the bends were chosen such that ε<10 and μ<10. It should be noted that an ideal plane wave with polarization perpendicular to the plane of propagation which is normally incident on the beam bend entrance surface will only interact with the radial component μr of the permeability tensor and the z-component εz of the permittivity tensor. In that case, the value of μϕ can be arbitrarily chosen.
Fig. 3. Distribution of the electric field component normal to the plane of propagation for a π/2-bend with (a) γ=0.02 (Media 1) and (b) γ=0.3 (Media 2). The dark grey lines represent the direction of power flow. The animation illustrates a phase advance of 2/9π per frame. Both realizations of a π/2-bend prove the reflectionless performance of the transformation-optical beam bend design independent on the choice of k.
To confirm the validity of this concept, we performed full wave simulations using Comsol as a finite element solver. In all numerical calculations the transformation-optical device was contained within a rectangular box with perfectly matched layer boundaries. The polarization of the electric field was chosen to lie in the z-direction normal to the (x-y)-plane of the wave propagation. The frequency of the wave used in the simulations was 8.5 GHz, however it should be noted that the transformation-optical approach is valid for any arbitrary frequency. The dark grey lines represent the direction of the power flow. Figure 3 shows the spatial distribution of the phasor of the electric field (color map) in the example of a π/2-bend. In Fig. 3(a) the material properties were calculated for γnorm=0.02 whereas in Fig. 3(b) the tensors were obtained for γnorm=0.3. As can be seen from Fig. 3, each material realization bends the incoming light by α=π/2. As a direct consequence of the different choices of γnorm in Eq. (16), the wavelength within the transformation-optical media differs for the various realizations. In the animation, the phase of the electric phasor is advanced by 2/9π per frame. In both cases, the waves propagate through the transformation-optical medium without reflections at the entrance or exit boundary to free space.
A similar behavior can be observed for radiation at oblique incidence. This is illustrated in Fig. 4 for incidence angles of Θ=π/8 (Fig. 4(a)) and Θ=2/9π (Fig. 4(b)). The π/2-bend at oblique incidence reveals the anisotropy of the transformation-optical medium in that the propagation direction of the wave fronts is almost normal to the direction of power flow. As a second interesting feature the angle between the incoming wave and the outgoing wave remains π/2 and is independent on the angle of incidence of the impinging radiation. Furthermore, no reflection could be observed at the entrance or exit surface of the transformation-optical beam bend at any incidence angle.
The concept of transformation optics becomes especially compelling in the context of reconfigurable metamaterials. Assume a metamaterial composed of sufficiently small unit cells, where the effective permittivity and permeability of each unit cell is tunable e. g. by applying a varying external voltage. In such a configuration, the electromagnetic properties of a metamaterial slab can be controlled in a way that an adaptive beam bend with tunable angle α between the incoming and the steered beam can be realized. Such adaptive transformation-optical devices based on tunable metamaterials would be of particular interest and almost unrivaled for the active manipulation of THz radiation where it is very difficult to actively control the electromagnetic fields by conventional optics. Voltage-controlled tunable metamaterials at THz frequencies were demonstrated by Padilla et al. [38]. The devices promise modulation frequencies of up to 1 MHz. A tunable α-bend can be realized by switching the values of the permittivity ε and permeability μ within a varying angular region of the medium between the vacuum values (εij=μij=δij) and the permittivity and permeability tensor values εij=μij of the α-bend (see Fig. 2 for typical values).
Fig. 4. (Movie) Distribution of the electric field component normal to the plane of propagation for a π/2-bend with γ=0.02 at oblique incidence at an angle (a) Θ=π/8 (Media 3) and (b) Θ=2/9π (Media 4). The dark grey lines represent the direction of power flow. The animation illustrates a phase advance of 2/9π per frame. The π/2-bend proves to operate reflectionless independently on the angle of incidence Θ.
Fig. 5. Distribution of the electric field component normal to the plane of propagation for (a) a π/9-bend with γ=0.05 (Media 5) and (b) for a π/3 beam splitter with γ=0.05 (Media 6). The dark grey lines represent the direction of power flow. In the animations, the angular domain where the effective material parameters are set by Eq. (16) is increased from α=0…3π/2 in steps of π/18 for the tunable beam bend in (a). In case of the beam splitter in (b) the angular region is changed from α=0…±3π/2 with a step size of ±π/18, where the positive sign applies to the left half and the negative sign to the right half of the splitter. The angular step size was arbitrarily chosen and can adopt any value. The phase of the electric field is advanced by 2/3π per frame.
Fig. 6. Two-dimensional intensity distribution (color map) for a (a) π/9-splitter, (b) 2/9π-splitter, (c) π/3-splitter, (d) 2/3π-splitter, (e) π/2-splitter and (f) π-splitter and the corresponding cross section plots of the intensity distribution along the white lines indicated in the color maps. The ratio between the intensity contained in the split beams and the intensity present in the scattered fields in the split gap increases with rising split angle 2α.
The potential of a dynamic metamaterial is illustrated in Fig. 5(a) in a full-wave simulation of an L-shaped metamaterial-based beam bend. By successively increasing and decreasing the angular region where the material properties of the beam bend are valid (see eqn. (16)), the incoming beam can be bent within an angle from 0 to 3/2π and vice versa. The wave propagation is simulated by advancing the phase of the electric field by 2/3π per frame in the animation of Fig. 5(a). The step size for the angular tuning of the beam bend was arbitrarily chosen to be π/18. The tunable transformation-optical beam bend operates without reflection at any bend angle. At bend angles α>π, the exiting wave from the bend interferes with the incoming wave which leads to the creation of continuously changing interference patterns. Note that the color scale is normalized to the maximum value of the electric field. This means, that at the time that constructive interference occurs at bend angles α>π, the magnitude of the electric field within the transformation-optical medium seems to be decreased in comparison to bend angles α≤π which is solely caused by the normalization process rather than being a physical consequence. In fact, the electric field strength inside the bend medium is equal for any angle at a given field strength of the incoming wave.
Fig. 7. Ratio between the maximum intensity in the split gap Igap and the maximum intensity in either the left or right branch of the split beams Isplit in dependence on the split angle 2α between the beams. At split angles 2α>π/2, the ratio is smaller than 4%.
As a further step, it is straightforward to realize a 2α-beam-splitter by merging a (+α)-bend and a (-α)-bend together. The simulation results are shown in Fig. 5(b) for the example of a tunable beam splitter. The phase of the electric phasor was advanced by 2/3π per frame and the split angle 2α was changed in a range from 0…3π with an arbitrarily chosen step size of π/9. As for the beam bends, the beam splitter operates without reflection independent of the angle 2α between the split beams. The scattered fields in the gap can be explained in terms of diffraction of the incoming waves at the splitting point. It should be noted that the impinging wave is diverging due to its finite width. For that reason, the observed split angle is larger than the angle 2α as calculated for a plane wave, since a diverging wave already possesses wave vector components (-kx) in the (-x)-direction and (+kx) in the (+x)-direction with respect to the point where the wave will finally be split. As a consequence, the observed split angle is larger for a diverging incoming wave than for a plane wave. The opposite would be the case for a converging incident wave on the beam splitter.
Figure 6 illustrates the two-dimensional intensity distributions (color maps) for various beam splitters with different split angles 2α. In addition, cross section plots of the intensity variation along a line normal to the propagation direction of the incoming beam at a fixed distance from the beam splitter, as indicated by the white lines in the color map graphs, are shown. As can be seen, the ratio between the intensity of the diffracted waves in the split gap and the intensity of the split beams decreases for increasing split angles 2α. This behavior is quantified in Fig. 7 where the ratio Igap/Isplit between the maximum intensity Igap in the split gap and the maximum intensity Isplit in either the left or right branch of the split beams is plotted in dependence on the split angle 2α. The ratio is rapidly decreasing for increasing split angles 2α and saturates for larger split angles. At 2α>π/2, the ratio Igap/Isplit is less than 0.04 which indicates that the beam splitter is working most efficiently in this range.
Finally, it should be mentioned that the concept of finite embedded coordinate transformations does not only apply to positive refractive materials. Applying the coordinate transformations for the beam bend to a negative refractive material with ε=μ=-1 instead of vacuum results in a beam bend with negative index of refraction. This is illustrated in Fig. 8 at the example of a π/2-bend with γ=0.02. The advance of the electric phasor clearly shows that the phase is propagating in the opposite direction of the power flow within the transformation-optical medium.
Fig. 8. (Movie) Distribution of the electric field component normal to the plane of propagation for a left-handed π/2-bend with γ=0.02 (Media 7). In the animation, the phase of the electric field is advanced by 2/9π per frame. The phase fronts are propagating in the opposite direction of the power flow inside the transformation-optical medium.
3. Transformation-optical beam expanders
In the following, we will describe the design of a specific type of a transformation-optical beam expander/compressor. In contrast to beam bends, this device obtained by the specific coordinate transformation inherently suffers from reflection loss when a pure transformation-optical approach is applied. This electromagnetic behavior can be explained in terms of a metric mismatch at the boundary between a transformation-optical expander and free space as we will discuss in the following.
One specific class of a transformation-optical expander/compressor consisting of a slab with thickness b can be described by
with the inverse transformation
where
and a is a parameter correlated to the focal spot size produced by the expander/compressor.
Calculating the Jacobi matrix of (17)-(19) and inserting in (1)-(2), one obtains with help of the inverse transformation (20)-(22) for the relative permittivity and permeability tensors
Fig. 9. Distribution of the electric field component normal to the plane of propagation for a beam expander with (a) a=6 and (b) a=-10. In the animation, the parameter a is changed from +6…-10 in steps of -2. The phase of the electric field is advanced by 2/9π per frame. For a≠0, reflections occur at the output facet of the lens which result in an amplitude modulation of the incoming wave. (Media 8)
The Jacobi matrix of (20)-(22) is
Starting from euclidian space in cartesian coordinates with the canonical basis {e1,e2,e3}, the basis vectors {e1′,e2′,e3′} of the transformed system can be obtained by use of ei′=Ai i′ ei and (25) as
In order to investigate the metric of the embedded transformation at the boundaries of the transformation-optical expander/compressor the projections 〈ei|ei′〉ei of the basis vectors {ei′} of the transformed system on the corresponding basis vectors {ei} of the untransformed space have to be calculated. This yields exactly the results already obtained in (26). In case of a metric matching at the boundaries the condition
has to be fulfilled. As can be seen from (26), (27) is fulfilled at x=-b/2 but violated at x=b/2. In this specific case the designed beam expander will inherently suffer from reflections occurring at the output face at x=b/2 due to an occurring impedance mismatch at the exit boundary.
To confirm the theoretical predictions, we performed full-wave simulations of a transformation-optical beam expander/compressor in Comsol. The wave frequency was arbitrarily chosen at 8.5 GHz. Figure 9(a) shows the electric field distribution of a beam expander with a=6. In the animation, the parameter a was changed from +6…-10 in steps of -2. The phase was advanced by 2/9π per frame. The animated wave propagation indicates that the incoming wave suffers from reflections at the exit surface of the transformation-optical component which results in an amplitude modulation of the waves propagating between the input port on the left side of the calculation domain and the exit boundary of the expander. Furthermore, the amplitude modulation depth is increasing for larger values of |a|, which is a direct consequence of the larger mismatch of the metrics between free space and the transformation-optical medium at higher values of |a| (see eqn. (26)). Apart from reflections, the simulations show that the beam expander/compressor provides well-predictable defocussing and focussing properties for incoming waves. The waves are focussed on the exit surface of the slab and diffract from there. At a parameter value a=-10 the strong focussing properties of the expander/compressor lead to the creation of an artificial point-like source (cf. Fig. 9(b)). As a main advantage of the transformation-optical approach it is possible to focus radiation beyond the diffraction limit inside the medium. On the other hand, the loss caused by reflections at the exit surface of the medium approaches considerable values for small focal spot sizes if a pure transformation-optical approach is applied. This shows, that there is a need in future work to combine transformation optics with conventional optical design methods, e. g. by designing anti-reflection coatings for transformation-optical components.
4. Conclusion
The methodology of finite embedded coordinate transformations was successfully employed for the design of adaptive beam bends, beam splitters and a beam expander. It was shown, that beam bends and splitters with arbitrary bend and split angles can be conceived by this approach. Furthermore, the beam bends and splitters were demonstrated to perform without reflection at the input and exit surfaces of the transformation-optical medium independent of the angle of incidence. It was explained how a reasonable design of beam bends and splitters can be derived from finite embedded coordinate transformations while ensuring moderate values of the relative permittivity and permeability values of transformation-optical bends and splitters with arbitrary bend and split angle at any given bend radius. The possibility to obtain moderate material properties makes transformation-optical beam bends and splitters especially suitable for fabrication. The beam bends and splitters were also discussed in the context of reconfigurable metamaterials. The adaptive behavior of bends and splitters with tunable bend and split angles was demonstrated in full wave simulations. Furthermore, a transformation-optical beam expander was presented. The analysis of a transformation-optical beam expander which was based on the compression or expansion of the coordinate space revealed that a reflectionless design is not possible following a pure transformation-optical approach. For this reason, a combination of conventional design methods for optical elements with the transformation-optical approach would be desirable, for example, in order to devise anti-reflection-coatings for transformation-optical components.
Acknowledgments
M. R. and D. R. S. acknowledge support from the Los Alamos National Laboratory LDRD program and a Multiple University Research Initiative sponsored by the Air Force Office of Scientific Research, Contract No. FA9550-06-1-0279.
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