Open Access
Add to BibSonomyAbstract
A flat reflector capable of scanning over wide angles is designed using a transformation optics approach. This reflector is derived from its virtual parabolic counterpart using a conformal coordinate transformation that determines the permittivity profile of the flat reflector. By changing the permittivity profile, the flat reflector is then capable of scanning up to 47° away from broadside while maintaining good beam characteristics across a wide frequency range. Moreover, its directivity is comparable to that of the virtual parabolic reflector, even at high scan angles. We use the Schwarz-Christoffel transformation as a versatile tool to produce perfect conformal mapping of coordinates between the virtual and flat reflectors, thereby avoiding the need to monitor the anisotropy of the material that results when employing quasi-conformal methods.
© 2013 Optical Society of America
1. Introduction
Transformation optics (TO) has been the topic of intense research in recent years as it provides a method for determining the full material tensors to manipulate electromagnetic waves. The basic principle of TO is intrinsically embedded in the metric invariance of Maxwell’s equations. Many interesting applications have emerged including cloaking and concealing objects [1–3], metamaterial lenses [4–6], wave collimators [7], light-guiding structures [8], beam bends and beam expanders [9]. TO also been used as the basis for designing antennas such as reflectors [10], high-directivity focusing antennas [10, 11] and steerable antennas [12]. However, since the electrical properties of the device are dictated by the derivatives of the coordinate transformation, the resulting permittivity and permeability tensors correspond to those of an anisotropic material. The need for a magnetic response and anisotropy in the permittivity and permeability tensors presents a significant challenge in realizing these devices. Consequently the majority of the TO devices realized to date possessing a magnetic response and anisotropy are limited to a particular polarization and have limited bandwidth [13–15].
Li and Pendry [3] have shown that an appropriate choice of coordinate transformation leads to isotropic materials in 2-D without the need for magnetic materials. Although the material is still inhomogeneous, it removes two of the most challenging aspects of material engineering –controllable permeability and anisotropy, especially at optical frequencies where the engineering of magnetic permeability is difficult. The resulting dielectric-only material is much easier to build compared to anisotropic magnetic materials. In addition, dielectric-only materials with relative permittivities larger than unity can be designed to operate away from resonances, which can dramatically increase the bandwidth of TO devices and potentially scale up to optical frequencies.
In this article, we propose to design a flat reflector that can be scanned by manipulating the dielectric properties of the materials covering the reflector, to mimic the mechanical scanning of a physical parabolic reflector. This dielectric-only reflector is optically the same as the traditional one but without a curved surface which can be geometrically advantageous. Theoretically this dielectric-only reflector can be used at any frequency as long as the required permittivity can be realized over the frequencies of interest. This work is part of a project whereby we wish to use electronically tunable materials to achieve beam-scanning, which will be initially pursued at microwave frequencies where such tunable materials can be more easily realized.
A flat reflector with occlusions can be used to manipulate the phase of the scattered field to shape the reflection or collimate it in specific directions. The phase change of the scattered field can be generated by changing the reflection coefficient on the surface of the reflector or altering the optical length experienced by the local field. In the former case, reflectarrays are often employed in which discrete tunable phase shifters are built into a planar surface [16–19]. In the latter case, the surface of reflector can be deformed to change the electrical path length experienced by the reflected wave. Such surfaces can be thought of as Fresnel zone reflectors [20–22].
The main goal this paper is to design a flat reflector to scan over as wide range of angles as possible over a broad frequency range. The flat reflector is designed by applying transformation optics techniques to a traditional parabolic reflector. This goal has been pursued using other TO techniques [23] but our work differs in some key aspects in the design of the reflector. Our virtual parabolic reflector is rotated about its apex instead of the center of the focal arc so that the reflector only needs to be rotated to half of the desired reflected angle. This method of rotation produces a much larger angle scanning capability which the flat reflector inherits. It will shown that the flat reflector can scan to at least 47° from broadside while maintaining excellent beam characteristics over a broad frequency range, which is significantly wider than scanning ranges reported in literature [23]. Moreover, the conformal coordinate transformation between the two reflectors is designed by using the Schwarz-Christoffel (SC) transformation which is a powerful and versatile conformal mapping technique that is rarely used in the context of transformation optics. Only a recent investigation [24] has applied the SC transformation to a phase modulator and a planar Luneburg lens. The SC transformation produces a perfectly conformal coordinate transformation between the two domains, thereby eliminating the need to monitor the degree of anisotropy that results in quasi-conformal methods [23].
2. Conformal coordinate transformation and Schwarz-Christoffel transformation
A coordinate transformation is illustrated in Fig. 1 where a conformal map f transforms a region in the z-plane to a region in the w-plane. The w- and z-plane are referred to as the virtual and physical spaces, respectively. Li and Pendry [3] have shown that when the map f is conformal, the resulting material in the physical space is isotropic and only possesses an electric response. This significantly simplifies the material design process and the material is much more suitable for realization. The transformation of permittivity is given by [25]:
where ε′r is the relatively permittivity in the z-plane and εr is the permittivity is the w-plane.
Fig. 1 An illustration of conformal mapping from z-plane, the physical space, to w-plane, the curved virtual space.
One of the many methods to accomplish a conformal transformation is by using a grid generation technique that is quasi-conformal. The resulting grid lines are all nearly orthogonal, creating a nearly conformal mapping between two domains. However, care must be exercised to ensure that the anisotropy is within acceptable limits [26]. Another method is to use the Riemann mapping theorem which says that a simple quadrilateral can be always conformally mapped to a rectangular region. The conformal map can be obtained by solving a set of Laplace’s equations [27] but one of the domains is restricted to be rectangular.
The Schwarz-Christoffel transformation is a conformal map between a canonical domain (the upper-half of the complex plane) to a simply-connected closed polygon. The closed polygon is useful for transformation optics devices because any simply-connected geometry can be approximated by using a series of points which define the vertices of the polygon. Although this conformal transformation has existed for some time, it has been rarely used as a technique to realize a conformal mapping in a transformation optics application. Only a recent investigation demonstrated the usefulness of such a versatile mapping technique [24]. The most basic SC transformation maps the upper-half of the complex plane to a closed polygon. However, a surprising number of variations of this coordinate transformation can be derived from this basic framework [28] and are very useful to generate conformal maps in transformation optics devices. Interested readers are encouraged to refer to textbooks [28, 29] for more details about the SC transformation. In the next section, we show three examples of SC transformations that aid in emulating a parabolic reflector with a flat one.
3. Reflector design using Schwarz-Christoffel transformation
3.1. Embedding a parabolic reflector in coordinate transformation
There are several ways to achieve beam-scanning using a reflector. Most commonly, the beam of a reflector can be scanned by laterally displacing the feed or the reflector, rotating the reflector about its focal point while feed remains fixed or rotating the entire feed and reflector assembly similar to a typical rotation of a satellite dish. In our application, we focus on a stationary feed that illuminates a tunable reflecting device to achieve wide angle beam-scanning without any mechanical movement.
Here we propose a new rotation mechanism in the virtual space shown in Fig. 2. A perfect electric conductor (PEC) in the shape of a parabola rotates about its apex, which is marked by a black circle. The dashed line denotes a reflector that has been rotated by an angle αo. A source feed is placed at the focal point of the reflector, which is shown in Fig. 2 as an open-ended rectangular waveguide (OEWG). As the reflector is rotated by an angle αo, the angle of the reflected beam is produced at an angle ϕo = 2αo. This method of rotation has the advantage over other rotation techniques because the reflector remains illuminated by the main lobe of the source as the reflector rotates to scan beams away from broadside. In addition, since the angle of rotation of the parabolic reflector only needs to be half of the angle of the reflected beam, it is expected that this method of rotation will give very good wide angle scanning performance.
To setup the SC transformation, the curved parabola is placed along a boundary of the coordinate transformation. Figure 3 shows three possible mapping in the w-plane (virtual space), whereby the blue lines denote the boundary of the SC transformation and hence the boundaries of the rectangular region in the z-plane (physical space). The portions of the boundary which are marked using red circles are sections of a parabolic reflector consisting of finite number of points which, together with rest of the vertices of the blue boundary, form a polygon. Figure 3(a) plots a mapping from the upper half of the complex plane to this open polygon shown here in the w-plane. The open polygon is useful in many TO applications such as carpet cloaks and beam shifters [3, 30], but our application focuses on transformation between finite-sized domains. Figure 3(b) shows the mapping from a rectangle in the z-plane to a closed polygon. The advantage of this closed polygon as opposed to the open polygon shown in Fig. 3(a) is that a well-defined boundary along y = 0 exists along which the coordinate transformation terminates. The parabolic portion of the boundary can be rotated about its apex to scan the beam as shown in Fig. 3(c). Note that the rectangular region is mapped to a different polygon for different angles of rotation.
Fig. 3 Three SC transformations mapped from different canonical regions – (a) mapping of the upper half of complex plane to an open polygon, (b) a mapping from a rectangle to a closed polygon which contains a parabolic section and (c) mapping from a rectangle to a close polygon which contains a parabolic section rotated by 20°.
Using the SC transformation is advantageous compared to traditional quasi-conformal mapping. Since the SC transformation is perfectly conformal, the resulting material properties are guaranteed to be perfectly isotropic with no approximation error. Recall from Eq. (1), the material permittivity in physical space is a scalar quantity. In quasi-conformal mapping, the degree of anisotropy must be checked at all points and evaluate whether anisotropy is within acceptable limits [23, 26]. This effect is especially pronounced when the geometry to be mapped has sharp corners. Figure 3(c) shows the rotated version of the parabolic reflector with a relatively sharp corner at the right edge of the reflector. The SC transformation handles the mapping with no anisotropy introduced.
4. Permittivity from coordinate transformation
4.1. Permittivity profile
The material permittivity is determined by the SC transformation according to Eq. (1). The parabolic reflector has a diameter of 5 λ at 5 GHz with an f/D ratio of 1.5. Figure 4(a) shows a SC transformation of the reflector rotated to 30°, which produces a reflected beam at 60° from broadside. A source feed is placed at the focal point of the reflector, which is shown as an open-ended waveguide fed with a point source. The entire transformation domain can be divided into three regions 1, 2, 3 as shown in Fig. 4(a), which map to regions A, B and C respectively in Fig. 4(b). The corresponding material permittivity is shown in Fig. 4(b) with two red markers showing the corresponding locations of the two edges of the PEC and a black circle marking the corresponding focal point. This flat reflector in physical space is equivalent to the virtual reflector in Fig. 4(a). Note that the physical size of the PEC region in Fig. 4(b) is slightly smaller compared to the PEC size in Fig. 4(a). A region D is also highlighted in dashed lines in Fig. 4(b), which will be explained shortly.
Fig. 4 SC transformation for a parabolic reflector with f/D = 1.5, D = 5 λ at 5 GHz. (a) virtual space, w-plane. (b) relative permittivity profile in physical space, z-plane.
There are some subtle but very important points regarding the permittivity profile that deserve special attention:
- This SC transformation maps a rectangle to a closed polygon. Hence the distance between any two points in the physical space z1 and z2 has the same optical length as the two corresponding points w1 = fsc(z1) and w2 = fsc(z2) in the virtual space. This is the direct result of the coordinate transformation. However, it does not guarantee the optical lengths are the same for two points if one of the points is outside of the transformation region. This poses a problem as the reflected fields extend beyond the transformed coordinate region, which can lead to de-focussing effects.
- Along the boundary of the coordinate transformation in the physical space, there is a impedance mismatch which would produce reflections at the boundary as the permittivity on one side of the boundary is that of free space while on the opposite side it is determined by |dw/dz|2. The reflections produced at these boundaries are expected to degrade the far-field pattern of the reflector.
- There is a degree of freedom in scaling the overall permittivity profile. If the coordinates of the rectangular region in Fig. 4(b) are divided by a factor of s = 2 to halve the physical size of the flat reflector, then the corresponding permittivity, according to Eq.(1), needs to be multiplied by a factor of s2 = 4. Hence the permittivity profile can be scaled if needed, providing a degree of freedom in the design of the reflector cover.
- There may be regions in the relative permittivity profile that are less than unity.
Here we discuss some techniques that mitigate the issues raised. It is clear in Fig. 4(b) that for majority of the area the permittivity does not change much. Hence, only region D is kept as the reflector cover. Note that if coordinate transformation could have been performed only in region D to start with instead of truncating the permittivity profile as shown in Fig. 4(b). However, it would take away a degree of freedom in choosing the desired permittivity values along the top of the reflector cover because the permittivity inside the transformation region is fixed by the coordinate transformation. The most intuitive and direct approach is to transform the entire region of space between the parabolic reflector and its focal point, as shown in Fig. 4(a), to ensure that this entire region is optically equivalent to the transformed domain in Fig. 4(b). The permittivity profile is then truncated to act as an approximation to the origin profile giving an extra degree of freedom in choosing the permittivity values at the top boundary of the reflector cover.
Next, consider the optical length experienced by a wave impinging on the physical reflector from the feed then reflecting back within the reflector cover and out into free space. This optical length must be the same as that the optical length experienced by the wave in the virtual space. Only then is the physical reflector optically equivalent to the virtual reflector. In this particular case of designing for a reflector, this amounts to making sure that the permittivity seen by the wave when transitioning from material to free space is smooth and free of reflections. It is worth mentioning here that in the case of designing for a lens, e.g. a traditional biconvex lens, researchers often perform coordinate transformation only on the lens itself [26] and do not perform coordinate transformation on the input and output side of the curved lens. Without coordinate transformation on the input and output side, a ray passing through a traditional curved-surface lens and passing through a flat version of the same lens have different optical lengths.
Regions 1 and 3 in Fig. 4(a) are designated as “extension regions” because they are included in the coordinate transformation but not directly associated with the reflector. The purpose of the extension regions is to provide ample room for the relative permittivity profile to approach unity near the boundaries of the transformation. Clearly, if the coordinate transformation is performed only in region 2, then there would be severe mismatch between the material boundary and free space, as evident in Fig. 4(b) where the relative permittivity profile along the left-most edge of region B is quite large. Hence, the presence of the extensions are important and their sizes need to be sufficiently large for the relative permittivity profile to taper down at the expense of a larger transformation optics device.
4.2. Reflector cover thickness and permittivity tapering
The thickness T of the flat reflector is chosen based on the desired maximum angle of scan from the flat reflector because for higher scan angles the peak permittivity is higher and it takes a thicker reflector in order for the relative permittivity profile to naturally taper down to unity to avoid reflections at the boundaries. Here we have been chosen a thickness T = 2.5 λ so that the relative permittivity is approximately 1.5 along the top of the reflector cover in Fig. 4(b). The sizes of the extension regions have been chosen to be le = 2.5 λ on each side to give sufficient room for the relative permittivity profile to decay down to a value close to unity.
This permittivity profile is for an extreme scan angle αo = 30° to produce a reflected beam directed at 60° from broadside. The maximum permittivity shown here is εr = 5 for visual clarity. There are 5 tapering regions in Fig. 5 highlighted in the dashed lines where the permittivity gradually tapers from the original permittivity profile shown in Fig. 4(b) to unity. The permittivity tapering in these regions is given by
where κ ∈ [0, 1] is a weighting parameter that depends on the distance from the boundary of the device. For example, along top of the reflector cover, κ is given by where T is the thickness of the reflector cover and Tt = λ/3 is thickness of the transition in the tapering region.Fig. 5 Permittivity profile for the flat reflector for αo = 30° for a reflected beam directed at ϕo = 60° from broadside.
5. Results
Since the realization of the flat reflector is expected at microwave frequencies first, we will assess the performance of the reflector using its far-field pattern at microwave frequencies. We choose a center frequency of 5 GHz so that the reflector diameter is 5 λ = 0.3 m and thickness of 2.5 λ = 150 mm. Simulations are setup in COMSOL Multiphysics 4.3 with a probe-fed OEWG placed at the focal point of the flat reflector. The aperture size of the waveguide is 60 mm which corresponds to a cut-off frequency of 2.5 GHz. Simulations have shown that the waveguide has multiple modes for frequencies greater than 7 GHz. The reason for using a rectangular waveguide as the source feed is that it can be easily realized and it has low back radiation. Hence, the frequency range of interest here is chosen to be 3 to 7 GHz centered at 5 GHz (an 80% relative bandwidth). Note that the frequency range is limited by the source feed and not by the reflector itself. However, the 80% bandwidth is sufficient to show the wideband operation of the reflector.
The far-field patterns are computed based on the scattered field of the reflector for the waveguide source. The 2-D directivity values reported here include the taper efficiency of the reflector but not the spillover efficiency because realizing a high spillover efficiency with minimal blockage would required very large reflector sizes, which would be computationally challenging. Hence, we consider spillover efficiency a secondary issue compared to demonstrating the operation of the flat reflector.
5.1. Scanning performance
Figure 6 shows directivity patterns of the flat parabolic reflector realized using permittivity profiles when the desired reflected beams angles at ϕo ∈ {0, 20, 40, 50, 60}°. The solid curves are the directivity patterns of the flat reflector and the dashed curves are of the parabolic reflector of the corresponding color. The corresponding relative permittivity profile are shown in Fig. 7 and Fig. 5 (for ϕo = 60°). The reflector cover has a thickness of 150 mm with no tapering and the regions of relative permittivity less than unity have been forced to 1 to avoid dispersive properties associated with medium with relative permittivity less than unity. Here we show the scanning angles only on one side but the reflector can scan to the opposite side by mirroring the permittivity profile.
Fig. 6 Directivity pattern at 5 GHz for the flat and the curved parabolic reflector for various scan angles ϕo ∈ {0, 20, 40, 50, 60}° shown as black radial lines. Solid curves are of the flat reflector and the dashed curves are of the curved parabolic reflector of the corresponding color.
Fig. 7 Relative permittivity profile for (a) ϕo = 0°, (b) ϕo = 20°, (c) ϕo = 40°, (d) ϕo = 50°. The permittivity profiles are truncated at a thickness of 2.5 λ. Note that color scale is limited to εr = 3 here to clearly show the relative permittivity distribution.
First, we examine the directivity patterns of the virtual parabolic reflector. These patterns are shown in dashed lines in Fig. 6. Clearly, the parabolic reflector produces an excellent scanning capability. Even at a large angle of 60° away from broadside, the directivity pattern does not significantly degrade from the pattern at broadside, despite the fact that the feed does not rotate. The overall shape of the pattern also remains approximately unchanged and the side lobe level is consistent at about −14 dB even at a scan angle of 60°. This justifies our choice of the rotation mechanism about the apex of the parabolic reflector. One reason for this good scanning performance is that the tilt angle of the reflector only needs to be half of the reflected beam while at the same time the reflector is not laterally moving compared to some of the other means of rotation that have been considered [23].
Second, we examine the directivity patterns of the flat reflector. These patterns are shown in solid lines in Fig. 6. Clearly, the flat reflector also has excellent scanning characteristics. Figure 6 shows that the flat reflector can scan up to 47° away from broadside despite the reflections at the boundaries even when regions with a relative permittivity less than unity have been disturbed. At the extreme scan angle of ϕo = 60°, the pattern of the flat reflector is degraded but it still has a main lobe at approximately 57°.
Figure 8 shows a plot of the actual scan angle of the reflected beam of the flat and parabolic reflectors and their maximum directivity versus the desired angle. The beam from the flat reflector is a few degrees off from the desired angle. This is due to the fact that regions of relative permittivity that are less than unity have had their relative permittivities forced to 1. The directivity from the flat reflector is at the highest value at broadside and it monotonically decreases for an increasing scan angle ϕo. This is expected behavior because the permittivity profile has a lower impedance mismatch at the device boundaries for lower scan angles and the regions exhibiting relative permittivities less than unity are smaller in area. When the flat reflector scans to 47° away from broadside, its directivity drops by 2.5 dB from the directivity at broadside and it is 2.9 dB lower compared to the directivity of the parabolic reflector scanned to the same angle. Here, we use a 3 dB criteria for the loss of directivity and we consider the 47° away from broadside as the maximum angle in which the flat reflector can scan.
Fig. 8 Angle of the actual reflected beam and maximum directivity as a function desired reflected beam angle ϕo. Solid lines are of the flat reflector and the dashed lines are of the parabolic reflector.
5.2. Effect of perturbing the relative permittivity profile
This section shows the effect of perturbing the permittivity profile on the directivity pattern. Figure 9 shows the directivity pattern for four cases, labeled A through D, of the permittivity profiles for a large scan angle of ϕo = 40° at a frequency of 5 GHz. Case A is when the permittivity profile is left undisturbed and no tapering exist along its boundaries. Case B is the same as the first except that the regions of relative permittivity less than 1 have been forced to have a relative permittivity of 1. Cases C and D are the same as the previous two cases respectively, except that permittivity is tapered as shown in Fig. 5. The directivity pattern for the parabolic reflector is also plotted for direct comparison.
The direction of the main lobes for all cases are approximately at the desired angle of 40°, which is drawn as a black radial line. Cases A and C each have their main lobes pointed closer to the desired angle and their main lobes are narrower compared to the main lobes of cases B and D, since they enjoy the luxury of the full permittivity range. This is expected behavior as the full permittivity range yields a better approximation to the curved reflector. However, the angle of the main lobe in cases B and D, where regions of less than unity permittivity are ignored, has not appreciably deviated from the desired scan angle, but the side lobe levels are higher than of that for cases A and C, leading to a lower directivity value. Here, the accuracy of the main lobe angle and the low side lobe levels are traded off for a much simpler material realization where no anisotropy and no magnetic response are needed. It is important to point out that since no materials with a relative permittivity less than unity are needed, no bandwidth limitations associated with the use of dispersive materials is imposed. Note that when the permittivity profile is tapered at the boundaries of the device, the directivity pattern has a slightly lower side lobe level compared the cases without the tapering region. One contributing factor is the smoother transition in the permittivity profile to that of free space. Overall, the flat reflector mimics the curved parabolic reflector quite well.
5.3. Directivity patterns versus frequency
Figure 10 shows the directivity pattern for a scan angle of ϕo = 40° over a range of frequencies. The solid curves are from the flat reflector and the dashed curves are from the curved parabolic reflector of the corresponding color. The permittivity profile used in the flat reflector is case B defined in the previous section. It is clearly shown in Fig. 10 that the directivity pattern of the reflector is very stable with frequency. The shape overall shape and the side lobe levels do not significantly change as the frequency is varied from 3 to 7 GHz.
Fig. 10 Directivity pattern for ϕo = 40° for various frequencies. Solid curves are from the flat reflector and the dashed curves are from the curved parabolic reflector of the corresponding color.
Figure 11(a) shows the maximum directivity as a function of frequency for various scan angles. The solid and dashed curves corresponds to that of the flat reflector and the curved parabolic reflector respectively. Figure 11 shows that the directivity decreases faster for the flat reflector for an increase in scan angle compared to the curved parabolic reflector. At broadside, the directivity of the flat and the curved reflector differs by about 1 dB on average across all frequencies. At ϕo = 50°, the directivities of the flat and the curved reflectors differ by about 2.5 dB on average. For a potentially tunable implementation of the flat reflector, this is an acceptable tradeoff for the ability to scan to such a large angle. At a scan angle of ϕo = 60°, the loss in directivity for the flat reflector becomes very significant, with an average of 5 dB loss compared to the directivity for the parabolic reflector. This is evident in Fig. 6 for a wide main lobe and high side lobe levels. At such extreme scan angles, the flat reflector does not have a similar behavior to the parabolic reflector.
Fig. 11 (a) Directivity as a function of frequency for various scan angles. (b) actual beam angle as a function of frequency. Solid curves and dashed curves are from the flat and curved parabolic reflector.
Figure 11(b) shows the actual angles of the main lobe from the flat and the parabolic reflector as a function of frequency. It is evident here that there exists a consistent pointing error in the angle of the main lobe from the flat reflector. This is caused by the disturbed permittivity profile where the regions of less than unit relative permittivity have been forced to 1. However, the main lobe from the flat reflector does not squint as frequency is varied from 3 to 7 GHz range indicating good wideband performance of the reflector, and such squinting could potentially be compensated for.
6. Practical considerations
It is desired that the overall size of the flat reflector and its feed source to remain as small as possible. Choosing a small f/D ratio keeps the feed source close to the flat reflector and it leads to an overall size reduction. However, the f/D ratio cannot be too small two reasons. First, a small f/D ratio yields parabolic reflectors with large curvature, which leads to a reflector cover in the physical space whose relative permittivity varies over a very large range. Second, the small f/D ratio leads to a permittivity profile that takes more room taper down to unity. Hence for the same truncation thickness for the reflector cover, the smaller f/D ratio would have more impedance mismatch at the boundary of the device which would negatively impact the beam characteristics of the reflector.
The performance of the reflector characterized in the paper has a continuous profile. In physical realization of the reflector cover, this continuous profile is difficult to build and it is much easier to realize a spatially discretized permittivity profile. It is reasonable to expect that using fewer discrete elements to approximate the continuous profile would lead to poorer reflector performance, due to the breakdown of the effective medium approximation. Moreover, realizing fewer elements would also lower the construction complexity and cost, especially when each of the elements potentially needs to be controlled individually. Therefore, a trade-off must be made between the performance of the flat reflector and its construction complexity. It is well known that a grid of sub-wavelength dipoles can produce an effective permittivity because they increase the effective polarization of the material. The dipoles can then be capacitively loaded to produce different effective permittivity. Fixed permittivity profiles have been realized for other transformation optics devices in the literature by using fixed dipole loadings [5, 31]. Variable capacitive loading [32] can also be realized by using varactor diodes at microwave frequencies. Varactor diodes can change their capacitance depending on the applied voltage. The realization of the tunable microwave realization of the flat reflector should be feasible.
7. Conclusion
A flat reflector is designed from a curved parabolic reflector using a TO approach. The use of the SC transformation produces a perfectly conformal coordinate mapping from the curved virtual parabolic reflector to the flat reflector. The SC transformation handles coordinate mappings even with relative sharp corners. The resulting relative permittivity profile is perfectly isotropic. Hence, there is no need to monitor the degree of anisotropy in the material properties compared to quasi-conformal coordinate transformation methods.
The virtual parabolic reflector is rotated about its apex to allow for an extremely large range of scan angles. The flat reflector inherits this feature and it is shown that it has good beam characteristics over a broad frequency range. The scan angle from the flat reflector is a few degrees away from the desired angle despite some impedance mismatch at the device boundaries and the fact that regions where the relative permittivity is less than unity have had their relative permittivities forced to 1. The error in the scan angle is relatively constant over a wide frequency range and the beam does not squint with frequency. Hence, the scan angle accuracy of the main lobe angle has been traded-off for a large bandwidth. Overall, the beam of the flat reflector tracks well with the ideal parabolic reflector as the beam is scanned away from broadside to 47°, at which point the directivity drops by 2.5 dB compared to broadside and the directivity is lower compared to an ideal parabolic reflector by 2.9 dB. Hence the flat reflector offers a broad 94° field of view. It was shown that the beam from the flat reflector has a very frequency-stable pattern from 3 to 7 GHz achieving an 80% relative bandwidth. Overall, the flat reflector closely mimics the beam characteristics of that of the curved parabolic reflector and it has a broad angle scanning range over a broad frequency range, making it a very attractive alternative to achieve large scan angles without the need to mechanically rotate the reflector.
References and links
1. T. Tyc and U. Leonhardt, “Broadband invisibility by non-euclidean cloaking,” Science 323, 110–112 (2009). [CrossRef]
2. R. Schmied, J. C. Halimeh, and M. Wegener, “Conformal carpet and grating cloaks,” Opt. Express 18, 24361–24367 (2010). [CrossRef] [PubMed]
3. J. Li and J. B. Pendry, “Hiding under the carpet: A new strategy for cloaking,” Phys. Rev. Lett. 101, 203901 (2008). [CrossRef] [PubMed]
4. D. Schurig, “An aberration-free lens with zero f-number,” New J. of Phys. 10, 115034 (2008). [CrossRef]
5. N. Kundtz and D. R. Smith, “Extreme-angle broadband metamaterial lens,” Nature Materials 9, 129 – 32 (2010). [CrossRef]
6. D. A. Roberts, N. Kundtz, and D. R. Smith, “Optical lens compression via transformation optics,” Opt. Express 17, 16535–16542 (2009). [CrossRef] [PubMed]
7. D. H. Kwon and D. H. Werner, “Transformation optical designs for wave collimators, flat lenses and right-angle bends,” New J. of Phys. 10, 115023 (2008). [CrossRef]
8. N. Engheta, “Antenna-guided light,” Science 21, 317–318 (2011). [CrossRef]
9. M. Rahm, D. A. Roberts, J. B. Pendry, and D. R. Smith, “Transformation-optical design of adaptive beam bends and beam expanders,” Opt. Express 16, 11555–11567 (2008). [CrossRef] [PubMed]
10. F. Kong, B.-I. Wu, J. A. Kong, J. Huangfu, S. Xi, and H. Chen, “Planar focusing antenna design by using coordinate transformation technology,” Appl. Phys. Lett. 91, 253509 –253509–3 (2007). [CrossRef]
11. P. H. Tichit, S. N. Burokur, and A. de Lustrac, “Ultradirective antenna via transformation optics,” J. Appl. Phys. 105, 104912 –104912–6 (2009). [CrossRef]
12. H. Chen, B.-I. Wu, L. Ran, T. M. Grzegorczyk, and J. A. Kong, “Controllable left-handed metamaterial and its application to a steerable antenna,” Appl. Phys. Lett. 89, 053509 (2006). [CrossRef]
13. Z. L. Mei and T. J. Cui, “Experimental realization of a broadband bend structure using gradient index metamaterials,” Opt. Express 17, 18354–18363 (2009). [CrossRef] [PubMed]
14. K. Aydin and E. Ozbay, “Capacitor-loaded split ring resonators as tunable metamaterial components,” J. Appl. Phys. 101, 024911 (2007). [CrossRef]
15. D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006). [CrossRef] [PubMed]
16. M. Riel and J. J. Laurin, “Design of an electronically beam scanning reflectarray using aperture-coupled elements,” IEEE Trans. Antennas Propag. 55, 1260 –1266 (2007). [CrossRef]
17. S. V. Hum, M. Okoniewski, and R. J. Davies, “Modeling and design of electronically tunable reflectarrays,” IEEE Trans. Antennas Propag. 55, 2200 –2210 (2007). [CrossRef]
18. M. Arrebola, J. A. Encinar, R. Cahill, and G. Toso, “Dual-reflector antenna with a reflectarray subreflector for wide beam scanning range at 120 GHz,” Int. Conf. Electromagn. in Advanced Applications , 848–851 (2012).
19. A. Gaebler, A. Moessinger, F. Goelden, A. Manabe, M. Goebel, R. Follmann, D. Koether, C. Modes, A. Kipka, M. Deckelmann, T. Rabe, B. Schulz, P. Kuchenbecker, A. Lapanik, S. Mueller, W. Haase, and R. Jakoby, “Liquid crystal-reconfigurable antenna concepts for space applications at microwave and millimeter waves,” Int. J. of Antennas Propag. (2009). [CrossRef]
20. L. Cabria, J. A. Garcia, J. Gutierrez-Rios, A. Tazon, and J. Vassal’lo, “Active reflectors: Possible solutions based on reflectarrays and Fresnel reflectors,” Int. J. Antennas Propag. (2009). [CrossRef]
21. J. Gutierrez-Rios and J. V. Sanz, “Simulated response of conic Fresnel zone plate reflectors (CFZPS),” in Europ. Conf. Antennas Propag. (2006). [CrossRef]
22. Y. Ji and M. Fujita, “Design and analysis of a folded Fresnel zone plate antenna,” Int. J. of Infrared and Millimeter Waves 15, 1385–1406 (1994). [CrossRef]
23. R. Yang, W. Tang, and Y. Hao, “Wideband beam-steerable flat reflectors via transformation optics,” IEEE Antennas Wireless Propag. Lett. 10, 1290 –1294 (2011). [CrossRef]
24. L. Tang, J. Yin, G. Yuan, J. Du, H. Gao, X. Dong, Y. Lu, and C. Du, “General conformal transformation method based on Schwarz-Christoffel approach,” Opt. Express 19, 15119–15126 (2011). [CrossRef] [PubMed]
25. U. Leonhardt, “Optical conformal mapping,” Science 23, 1777–1780 (2006). [CrossRef]
26. W. Tang, C. Argyropoulos, E. Kallos, W. Song, and Y. Hao, “Discrete coordinate transformation for designing all-dielectric flat antennas,” IEEE Trans. Antennas Propag. 58, 3795 –3804 (2010). [CrossRef]
27. Y. G. Ma, N. Wang, and C. K. Ong, “Application of inverse, strict conformal transformation to design waveguide devices,” J. Opt. Soc. Am. A 27, 968–972 (2010). [CrossRef]
28. T. A. Driscoll, A MATLAB toolbox for Schwartz-Christoffel mapping (ACM Trans. Math. Softw., 1996).
29. D. Wunsch, Complex Variables with Applications (Addison Wesley, 1993).
30. N. Kundtz, D. R. Smith, and J. B. Pendry, “Electromagnetic design with transformation optics,” Proceedings of the IEEE 99, 1622 –1633 (2011). [CrossRef]
31. J. P. Turpin, Z. H. Jiang, P. L. Werner, and D. H. Werner, “Tunable metamaterials for conformally mapped transformation optics lenses,” IEEE Proc. AP–S Int. Symp. Antennas Propag. (2010).
32. S. V. Hum, M. Okoniewski, and R. J. Davies, “Realizing an electronically tunable reflectarray using varactor diode-tuned elements,” IEEE Microw. Wireless Compon. Lett. (2005). [CrossRef]
