## Abstract

The past decade has seen a revolution in electromagnetics due to the development of metamaterials. These artificial composites have been fashioned to exhibit exotic effects such as a negative index of refraction. However, the full potential of metamaterial devices has only been hinted at. By combining metameterials with transformation optics (TO), researchers have demonstrated an invisibility cloak. Subsequently, quasi-conformal mapping was used to create a device that exhibited a broadband cloaking effect. Here we extend this latter approach to a strictly conformal mapping to create reflection less, inherently isotropic, and broadband photonic devices. Our method combines the novel effects of TO with the practicality of all-dielectric construction. We show that our structures are capable of guiding light in an almost arbitrary fashion over an unprecedented range of frequencies.

©2009 Optical Society of America

Transformation optics yields a precise prescription of spatially-varying and anisotropic optical constants to guide electromagnetic radiation. TO accomplishes this feat by taking advantage of the invariance of Maxwell's Equations upon coordinate transformations. The TO methodology was initially applied only to the design of electromagnetic cloaks [1,2], since coordinate transformations of free space invariably returned waves undisturbed as they existed the altered region.

Later works demonstrated that the TO methodology could be applied to create devices that could steer or focus electromagnetic waves [3]. This class of transformations are termed finite-embedded transformations, and this led to the design of various optical components such as beam and waveguide bends [4,5], shifters, splitters, and imaging elements [6].

Unfortunately, very few of these transformations lend themselves to easily realizable devices. Traditional TO designs require control of the tensor components of both the electric permittivity ε and magnetic permeability μ of constituent materials. Even though this control is possible by virtue of electromagnetic metamaterials [7–10], it severely complicates the design process. Additionally, the use of highly-dispersive resonant elements, i.e. $\epsilon =\epsilon \left(\omega \right)$ and $\mu =\mu \left(\omega \right)$, limits the operation of devices to a single frequency. Various restrictions and approximations [11,12] somewhat alleviate this complexity, but they also reduce the performance of potential photonic devices.

More recently Li and Pendry conjectured that the design complexities associated with TO could be minimized by an appropriate choice of coordinate transformation [13]. Specifically, by numerically computing the quasi-conformal map of a Cartesian grid to a slightly deformed geometry, they showed that a cloaking effect could be created. The quasi-conformal mapping minimized the anisotropy needed in the optical constants, which meant that the cloak could be approximated by an isotropic dielectric material, (although still spatially varying), with negligible degradation in performance. This design was subsequently demonstrated at microwave frequencies [14], and more recently at optical [15,16].

The quasi-conformal mapping technique is a powerful tool and may result in the successful demonstration of many practical devices. However, we demonstrate that if the virtual domain is not chosen judiciously, this technique is not practical for many photonic applications. Rather, we use our control over the virtual domain to employ a strictly conformal mapping method [17] and create broadband, all-dielectric photonic devices. We demonstrate advantages this method achieves over existing quasi-conformal or analytic techniques, by describing a class of conformal mappings, which could be utilized as waveguide beam bends, splitters, or couplers.

We begin by considering a rectangular (virtual) domain that is mapped to some distorted (physical) domain, as shown in Fig. 1
. We first overlay the virtual domain with a square grid consisting of $a\times b$ cells of side length *δ*. The quasi-conformal technique will transform each square $\delta \times \delta $ cell in the virtual domain to a rectangle in the physical domain. The aspect ratio *α* of the rectangular cell will be constant for all cells in the transformed space, and is given by,

*M*is the conformal module of the physical domain and

*m*is that of the virtual domain, and one selects either +1 or −1, whichever yields the larger value for

*α*. The conformal module is the unique value of

*h/w*(Fig. 1) for which the quadrilateral

*Q*, shown in Fig. 1, is conformally equivalent to the rectangular quadrilateral

*R*such that for

*h/w = m*and only for this value there exists a unique conformal mapping that takes the four points {A',B',C',D'} respectively onto the four vertices {A,B,C,D} of

*R*in a counter-clockwise order. For the rectangular virtual domain

*R*, shown in Fig. 1, the conformal module is clearly

The anisotropy factor *α* presented in Eq. (1) can also be given in terms of the metric tensor *g* of the transformation,

In the quasi-conformal mapping technique the anisotropy factor plus its inverse is minimized, bringing the anisotropy factor as close to unity as possible. However, depending on the ratio of modules, this optimization may yield an anisotropy factor that is substantially different than one. For the cloaking application recently discussed in the literature, the module ratio happens to be somewhat close to unity and is accurately approximated by one. For Transverse Electric (TE) incidence, this allows the approximation

while the permittivity is given bywhere*ε*is the dielectric constant,

*μ*is the permeability,

*ε*is the background dielectric, and

_{r}*μ*is the background permeability. Typically, both

_{r}*M*and the map itself must be determined numerically [18,19]. We begin by determining the quasiconformal map and

*M*, which allows us to calculate the conformal map.

Consider the rectangular region R of space in Fig. 1 to be spanned by the orthogonal coordinates $\left(\xi ,\eta \right)$ such that $0\le \xi \le 1$ and $0\le \eta \le M$. A transformed coordinate system that maps R to Q can be defined as $\colorbox[rgb]{}{$x$}=\left(x,y\right)=\left(x\left(\xi ,\eta \right),x\left(\xi ,\eta \right)\right)$. The physical domain is defined by the four arcs ${\colorbox[rgb]{}{$x$}}_{l}\left(\eta \right)$, ${\colorbox[rgb]{}{$x$}}_{r}\left(\eta \right)$, ${\colorbox[rgb]{}{$x$}}_{b}\left(\xi \right)$, and ${\colorbox[rgb]{}{$x$}}_{t}\left(\xi \right)$. The conformal map satisfies the Cauchy-Riemann equations everywhere in the domain:

*m*is unknown, we cannot use (5) to calculate the conformal map. Rather, we must calculate the quasiconformal map by solving the Beltrami equations,

*f*is constant and invoking the equality of mixed partial derivatives, Eq. (7) may be rewritten as

*f*is known, these equations become a set of linear, elliptical partial differential equations that can be discretized and solved

*via*a relaxation method. However, if

*f*is unknown, these equations are nonlinear and

*m*must be calculated iteratively according to

Additionally, the boundary points must be updated to guarantee orthogonality. This condition is expressed as

The limitation of quasi-conformal mapping stems from the fact that, as mentioned, *α* also represents the anisotropy of the material. Therefore, if *M/m* is not approximately unity, the device cannot be made isotropic, and magnetic coupling, i.e. $\mu \ne 1$, will in general be necessary. This will often be the case if the virtual domain is chosen without regard to its module and in some cases where the virtual domain is constrained to have a particular and unfavorable module by the application.

We consider here two dimensional two-port devices, where waves can enter and exit through the device boundary along two separate segments - such as the case for waveguides. In addition to the shapes of the virtual and physical domain boundaries, the mapping for these devices requires specification of the mapping of the four points that bound the two ports. These requirements over-specify the map. The most general form of the Riemann mapping theorem does not apply and a conformal map does not, in general, exist that satisfies the specifications. However, if the virtual and physical domains have the same module, it can be shown that a conformal map does exist [20]. Specifically, we assign the segments of length *w* to be the input and output facets of a waveguide, and choose *w* to be the same in both domains. We are thus free to let *h* vary in such a way that a strictly conformal map exists. In this fashion the *w* boundaries can be made reflection less – determined by the metric of the transformation *g* at the interface [21]. For finite, embedded mappings, *g* must be expressed in a basis consisting of local coordinates parallel and orthogonal to the port boundaries. If *g* is equal to the free-space metric, then no reflections will occur on the boundary. Notably, for the conformal map, Eq. (5) becomes exact and we can write

*g*) must be that of free space.

We now turn to demonstration of photonic devices designed with the conformal TO methods described here. All of the following discussion assumes that waveguide bends are bounded by PEC sidewalls. Figure 2(a)
shows the spatially varying relative permittivity *ε/ε _{r}* for a circular beam or waveguide bend, determined computationally by the conformal module of Eq. (1) and specified by Eq. (6). The permittivity has a maximum value of 3.25 and a minimum value of 0.5. The index of refraction ($n=\sqrt{\epsilon /{\epsilon}_{r}}$) is higher on the inner curve and causes the wavefronts to lag those on the outer curve, forcing the beam to bend as shown in Fig. 2(b).

The smooth boundaries employed in Fig. 2(a) yield a smaller range of the index of refraction than would sharp corners. However, it should be noted that the algorithm can be applied to any general polygon. For example, consider a sharp, 90° waveguide corner in an empty, lossless waveguide. Such a corner will, of course, cause all incident power to be reflected back to the source. However, by filling the space with a conformally-mapped spatially varying dielectric (as depicted in Fig. 2(c)), we can eliminate reflections and guide the wave smoothly around the corner, as shown Fig. 2(d). The discretized permittivity map used in the simulation in Fig. 2(d) had a maximum of value *ε*= 30*ε _{r}* at the inside corner.

We emphasize that this algorithm is not limited to the two cases described above. The Riemann mapping theorem guarantees that any generalized quadrilateral can be mapped by a rectangle of the same conformal module. To illustrate this generality, we show a “squeezed” waveguide bend in Fig. 3 . Using these conformal methods, the map could also be constructed to guide waves around deformations in the wall of the guide, so that the propagation is similar to [13]. This method can therefore be use as a broadband substitute for epsilon-near-zero (ENZ) tunneling [22–26], which is inherently narrowband. Instead of increasing the spatial wavelength of the wave by decreasing the refractive index to near zero, our method uses a spatially-varying index to guide the wave around deformities and bends in the guide. Additionally, complicated impedance matching techniques are unnecessary as the transformed region can be constructed to match at the boundary. The advantage of ENZ tunnelling is that the method is independent of the tunnelling domain, unlike in our device where the material properties must be calculated for each unique physical geometry. The conformal mapping technique could also be applied to create a form of directional concentrator [27]. Collimated light entering the device could be concentrated by shaping the device in a manner similar to the one demonstrated in Fig. 3. Power is concentrated in the squeezed region.

Finally, as an extreme case, we show that waves could be constrained to follow a complicated arbitrary elongated path, and the results are visually striking. A straight rectangular waveguide section is mapped to the Eastern Seaboard of the United States, left portion of Fig. 4 . As shown in the right portion of Fig. 4 the wave smoothly propagates within the guide with little noticeable distortion or degradation. Around Maine, the waveguide is compressed to below cutoff for TE modes of the empty waveguide, and no propagation can occur without the TO medium (Fig. 4(b)). Complicated maps such as these do not present a barrier to device construction utilizing the computational conformal transformations outlined here, since exotic material parameters are unnecessary. Notably, the permittivity depends only on the local distortion of the grid lines, and not complexity of the map itself.

We next highlight the salient features of the TO method for the construction of novel photonic devices. The method described here is conformal. Thus one can always determine the unique mapping that permits various photonic devices to be constructed, without the necessity of anisotropic, resonant metamaterials. For example, if we wanted to fabricate the waveguide, shown in Figs. 2(a) and 2(b), in free space then we would be forced to use resonant metamaterials in order to obtain dielectric values less than unity. As with all TO methods, the material prescription will often involve material properties with values less than those of free space, requiring a resonant, and thus lossy, response. This can often be mitigated if the application allows a free choosing of the background medium. Thus this bend could be made without resonant narrowband metamaterials by embedding it in a dielectric substrate $\epsilon =6.5{\epsilon}_{r}.$ In this manner the bend will yield broadband low loss performance, so long as the embedding dielectric varies slowly over the desired operational wavelength.

Another impressive property of the conformal mapping demonstrated here, is that the physical domain that results is completely equivalent to the virtual domain. Thus no matter how complicated the mapping, the distorted space retains the properties of a 2D straight waveguide segment. These 2D results are also extensible to physically realizable 3D waveguides for the TEn0 guide modes. For example, consider conventional plasmon [28] and photonic dielectric waveguide bends [29]. Normally, one would incur radiative losses due to the bend and thus a minimum bend radius can be achieved without significantly reducing performance. However, conformally TO designed dielectric photonic waveguide bends can be made with arbitrarily small bend radii with *no resonant losses*, since the transformed region is equivalent to a straight waveguide segment. Conformally-mapped regions also present an important simplification to subsequent calculations involving the region, as propagation through the mapped region will be identical to propagation through a straight region of length *h* in the virtual domain. One may also fabricate miniaturized photonic waveguide components. For example using ‘squeezed’ waveguides, as shown in Fig. 3, we may compress the photonic waveguide, while constructing the bends, splitters, and couplers that constitute the entire photonic circuit.

We have presented a conformal mapping method which permits the construction of various easily realizable photonic devices. The method illustrates that conformally-mapped quadrilaterals can effectively distort guided light or collimated beams. Specific examples of waveguide bends, a concentrator, and a complicated beam bend have been demonstrated.

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