Download Guiding light with conformal transformations

Guiding light with conformal transformations
Nathan I. Landy and Willie J. Padilla
Department of Physics, Boston College, 140 Commonwealth Avenue, Chestnut Hill, MA 02467, USA
*[email protected]
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
OCIS codes: (160.3918) Metamaterials; (230.7370) Waveguides.
References and links
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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. ε = ε ( ω )
and µ = µ ( ω ) , 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
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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.
Fig. 1. Schematic detailing the conformal transformation optical method. A rectangle can be
mapped conformally to any “quadrilateral” of the same conformal module M.
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 × b cells of side length δ. The quasi-conformal technique will transform each
square δ × δ 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,
±1
 M  
 ,
 m  
α = max 
(1)
where 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
 h ±1 
m = max    .
(2)
 w  
The anisotropy factor α presented in Eq. (1) can also be given in terms of the metric tensor
g of the transformation,
α + α −1 =
Tr(g )
(3)
.
det( g )
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
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µ / µ r ≈ 1,
(4)
ε / ε r = (det g ) −1/ 2 .
(5)
while the permittivity is given by
where ε is the dielectric constant, µ is the permeability, εr is the background dielectric, and µr
is the background permeability. Typically, both 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 (ξ ,η ) such that 0 ≤ ξ ≤ 1 and 0 ≤ η ≤ M . A transformed coordinate system that
maps R to Q can be defined as x = ( x, y ) = ( x (ξ ,η ) , x (ξ ,η ) ) . The physical domain is defined
by the four arcs xl (η ) , x r (η ) , xb (ξ ) , and xt (ξ ) . The conformal map satisfies the CauchyRiemann equations everywhere in the domain:
xξ = yη ,
yξ = − xη .
(6)
Since m is unknown, we cannot use (5) to calculate the conformal map. Rather, we must
calculate the quasiconformal map by solving the Beltrami equations,
fxξ = yη ,
fyξ = − xη .
(7)
where f = M / m . By noting that f is constant and invoking the equality of mixed partial
derivatives, Eq. (7) may be rewritten as
f 2 xξξ + xηη = 0,
f 2 yξξ + yηη = 0.
(8)
If 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
f2 =
∫∫ fdξ dη .
∫∫ f dξ dη
−1
(9)
Additionally, the boundary points must be updated to guarantee orthogonality. This
condition is expressed as
xξ ⋅ xη = 0 on ξ = 0,1 and η = 0, m.
(10)
This condition can be discretized in terms of the interior points and used to solve for the
parameter values along the boundary curves that satisfy the orthogonality condition. Once the
iterative process has converged, we make the substitution f −1dη → dη or fd ξ → d ξ to give
us the conformal map.
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. µ ≠ 1 , will in general be
necessary. This will often be the case if the virtual domain is chosen without regard to its
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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
n / nr = (det g ) −1/ 4 ,
(11)
and further this condition is simplified such that only the basis-independent quantity det(g)
must be that of free space.
Fig. 2. Permittivity map and simulated transmitted electric field of conformally mapped
waveguides. (a) and (b) have rounded corners whereas (c) and (d) have straight corners. The
upper color bar represents the range of the permittivity values in the maps in (a) and (c). The
lower bar represents the electric field values in (b) and (d)
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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/εr
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 = ε / ε 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= 30εr 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
Fig. 3. Permittivity map and simulated transmitted electric field of a ‘squeezed’ bend. (a) is the
permittivity map and permittivity color range. (b) shows simulated wave propagation and the
normalized electric field color map.
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
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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.
Fig. 4. Guided waves around the eastern United States. (a) shows the permittivity and
conformal map. The range of permittivity values and color map are shown on the right of (a).
(b) shows wave propagation in the guide without the permittivity map. (c) shows wave
propagation with the map. The color map for the normalized electric field is shown on the right
of (c). an animation of the wave propagation in (b) and (c) is provided in Media 1.
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
ε = 6.5ε 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
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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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