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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 22, Iss. 15 — Jul. 28, 2014
  • pp: 18490–18498
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Anisotropy minimization via least squares method for transformation optics

Mateus A. F. C. Junqueira, Lucas H. Gabrielli, and Danilo H. Spadoti  »View Author Affiliations


Optics Express, Vol. 22, Issue 15, pp. 18490-18498 (2014)
http://dx.doi.org/10.1364/OE.22.018490


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Abstract

In this work the least squares method is used to reduce anisotropy in transformation optics technique. To apply the least squares method a power series is added on the coordinate transformation functions. The series coefficients were calculated to reduce the deviations in Cauchy-Riemann equations, which, when satisfied, result in both conformal transformations and isotropic media. We also present a mathematical treatment for the special case of transformation optics to design waveguides. To demonstrate the proposed technique a waveguide with a 30° of bend and with a 50% of increase in its output width was designed. The results show that our technique is simultaneously straightforward to be implement and effective in reducing the anisotropy of the transformation for an extremely low value close to zero.

© 2014 Optical Society of America

1. Introduction

In recent years, the transformation optics (TO) technique has attracted the attention of researchers due to the great freedom that it allows in designing electromagnetic devices with unconventional functionalities. To make use of TO, coordinate transformations describing the desired functionality is necessary. TO then maps that transformation to the actual non-medium that will make the final electromagnetic device. However, the media obtained via TO are, in general, both inhomogeneous and anisotropic, which, renders them difficult to obtain in practice. The former is a fundamental part of TO, but to overcome the latter, researchers have proposed the use of quasi-conformal mappings [1

1. L. Ahlfors, “On quasi-conformal mappings,” J. Anal. Math. 3(1), 1–58 (1953). [CrossRef]

], which minimizes the anisotropy [2

2. J. Li and J. B. Pendry, “Hiding under the carpet: A new strategy for cloaking,” Phys. Rev. Lett. 101(20), 203901 (2008). [CrossRef] [PubMed]

] in the original coordinate transformation.

This paper introduces a new technique of quasi-conformal mapping for transformation optics with the application of the least squares method (LSM) to reduce anisotropy. To this end, perturbation functions in the form of power series are added to the coordinate transformation. The power series coefficients can be optimized via LSM to algebraically design electromagnetic media with extremely low anisotropy while meeting all boundary conditions specified in the initial design. Finally, we present our technique particularized for the treatment of waveguide-based photonic devices, and demonstrate it by designing a 30° bend with simultaneous width scaling.

2. Theoretical development

According to Fermat's physical principle, a ray of light travels through the shortest optical path between two points. If on the one hand this principle allows one to calculate the path traveled by light from the knowledge of the refractive index distribution of the environment, on the other hand it does not permit the derivation of a suitable media based on prescribed optical paths. This inverse problem is what TO aims to solve, and not only in the ray optics domain, but also in the full vectorial electromagnetic field model.

2.1 Transformation optics

The TO theory is based on the fact that Maxwell's equations are invariant to coordinate transformations. It has been shown that the space deformation imposed by a coordinate transformation is maps the path traveled by light in a corresponding medium [3

3. U. Leonhardt and T. G. Philbin, “Transformation optics and the geometry of light,” Prog. Opt. 53, 69–152 (2009). [CrossRef]

] with permittivity and permeability tensors defined by the transformation itself.

Considering the transformation from a cartesian system, the electric permittivity, ε, magnetic permeability, μ, of transformed medium are obtained from [3

3. U. Leonhardt and T. G. Philbin, “Transformation optics and the geometry of light,” Prog. Opt. 53, 69–152 (2009). [CrossRef]

]:
εij=εggij
(1)
μij=μggij
(2)
where, gij is the metric tensor of the coordinate transformation and g is its determinant.

2.2 Coordinate transformation

For a two-dimensional coordinate system, one can define a coordinate transformation (CT) through the use of a complex function w: (real and imaginary parts represent the coordinates in each dimension). If a complex function w(z) is differentiable in D, then w(z) is said to be analytic in D. The function is analytic if, and only if, the Cauchy-Riemann conditions are satisfied. In turn, the coordinate transformation represented by w(z) is said to be conformal [4

4. G. E. Shilov, Elementary Real and Complex Analysis (MIT, 1973).

].

In TO, a conformal transformation results in an isotropic medium [3

3. U. Leonhardt and T. G. Philbin, “Transformation optics and the geometry of light,” Prog. Opt. 53, 69–152 (2009). [CrossRef]

], a desirable characteristic to simplify the fabrication of the final device.

These conditions cannot be met by a conformal transformation (except the identity), as discussed in [6

6. U. Leonhardt, “Optical conformal mapping,” Science 312(5781), 1777–1780 (2006). [CrossRef] [PubMed]

]. This follows from the uniqueness theorem [4, Theorem 10.39] from complex analysis, which states that if a conformal transformation coincides with the original system in a certain region, then the transformation and the original systems must coincide at all points. Therefore, the use of a conformal transformation for the benefit of an isotropic medium will unavoidably result in reflections at the interfaces as well.

Alternatively, one can use quasi-conforming mappings [1

1. L. Ahlfors, “On quasi-conformal mappings,” J. Anal. Math. 3(1), 1–58 (1953). [CrossRef]

] (mappings presenting reduced anisotropy [2

2. J. Li and J. B. Pendry, “Hiding under the carpet: A new strategy for cloaking,” Phys. Rev. Lett. 101(20), 203901 (2008). [CrossRef] [PubMed]

]) that may or not meet the boundary conditions to avoid reflections. These mappings have small deviations from the Cauchy-Riemann conditions [1

1. L. Ahlfors, “On quasi-conformal mappings,” J. Anal. Math. 3(1), 1–58 (1953). [CrossRef]

]. For example, the well-known quasi-conformal technique proposed by Li and Pendry [2

2. J. Li and J. B. Pendry, “Hiding under the carpet: A new strategy for cloaking,” Phys. Rev. Lett. 101(20), 203901 (2008). [CrossRef] [PubMed]

] effectively minimizes the average anisotropy, but due to its sliding boundary conditions, the technique end up generating reflections at the edges of the transformed medium.

Chang et al. [7

7. Z. Chang, X. Zhou, J. Hu, and G. Hu, “Design method for quasi-isotropic transformation materials based on inverse Laplace’s equation with sliding boundaries,” Opt. Express 18(6), 6089–6096 (2010). [CrossRef] [PubMed]

], in a different way from the one in [2

2. J. Li and J. B. Pendry, “Hiding under the carpet: A new strategy for cloaking,” Phys. Rev. Lett. 101(20), 203901 (2008). [CrossRef] [PubMed]

], also proposed a strategy to reduce the anisotropy, but still with the same disadvantage of using sliding boundary conditions. Other variations for reducing anisotropy via slipping boundary conditions can be found in [8

8. N. I. Landy and W. J. Padilla, “Guiding light with conformal transformations,” Opt. Express 17(17), 14872–14879 (2009). [CrossRef] [PubMed]

] and [9

9. 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(5), 968–972 (2010). [CrossRef] [PubMed]

]. These works demonstrated that quasi-conformal mapping can be used to perform practical devices such as lens, bends and expanders [7

7. Z. Chang, X. Zhou, J. Hu, and G. Hu, “Design method for quasi-isotropic transformation materials based on inverse Laplace’s equation with sliding boundaries,” Opt. Express 18(6), 6089–6096 (2010). [CrossRef] [PubMed]

13

13. D. Liu, L. H. Gabrielli, M. Lipson, and S. G. Johnson, “Transformation inverse design,” Opt. Express 21(12), 14223–14243 (2013). [CrossRef] [PubMed]

].

Although the mentioned techniques of quasi-conformal are 2D mapping, it can be easily extend to 3D by extruding or rotating the obtained refractive index profile. In [11

11. H. F. Ma and T. J. Cui, “Three-dimensional broadband and broad-angle transformation-optics lens,” Nat. Commun. 1, 124 (2010).

] is presented a three-dimensional broadband and broad-angle lens by rotating an isotropic refractive index profile.

Once a quasi-conformal mapping is obtained, we can assume the medium is isotropic by ignoring the residual anisotropy. As presented in [14

14. D. H. Spadoti, L. H. Gabrielli, C. B. Poitras, and M. Lipson, “Focusing light in a curved-space,” Opt. Express 18(3), 3181–3186 (2010). [CrossRef] [PubMed]

], the refractive index distribution may be implemented, for example, with a pattern of holes (air, n = 1.0) in a silicon (n = 3.5) substrate for integration with conventional silicon photonics devices.

To quantify the residual anisotropy in the transformed medium we use the measure K defined below [15

15. K. Astala, T. Iwaniec, and G. Martin, Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane (Princeton University, 2008).

]:
K=tr(JTJ)2det(J)
(3)
where, J is Jacobian matrix of the CT. Assuming a quasi-conformal transformation, the isotropic index of refraction for the resulting medium can be approximated by:

n2=n2tr(g2gij)
(4)

3. Mathematical development

The coordinate transformation defined by (5) and (6) has the advantage of being general, allowing its application for any transformation (with adequate choice for b(x,y)) and, automatically, absorbing the boundary conditions (facilitating the optimization) without losing of simplicity.

In TO, the boundary conditions usually determine the functionality of the desired optical devices. Therefore, it is of paramount importance that the anisotropy minimization technique does not change the boundary conditions for the engineered transformation.

3.1 Transformation optics for planar waveguides

Although the proposed technique is general, we will give special attention to the case where TO is used to design waveguide-based devices. As previously mentioned, the term b(x,y)must vanish at the device boundaries to avoid reflections. This condition alone is, however, not sufficient. To ensure that the waveguide and/or the electromagnetic wave fronts are planar at the input and output boundaries we must also satisfy:

b(x,y)x|bnd=0orb(x,y)y|bnd=0
(7)

Fig. 1 General transformation applied to a y-directed waveguide. A LSM optimizes the transformation with sample points chosen iteratively.
Assuming these conditions are met for the case of a waveguide with the propagation along the y-axis in the original medium (see Fig. 1), the wave propagation vector at the output of the transformed medium will have angle θ relative to the x'-axis such that:

tg(θ)=yyxy=fyyfxy
(8)

We also define φ, the device output boundary angle:

tg(φ)=yxxx=fyxfxx
(9)

Figure 1 illustrates the physical meaning of angles and in the transformed structure.

If the propagating angle θ is not constant over the guide output facet, the electromagnetic wave fronts exiting the device are not planar. Similarly, if the angle φ is not constant, the waveguide output itself is not flat.

3.2 Application of the least squares method

Once the coordinate transformation is defined according to expressions (5) and (6), we use the LSM to determine the coefficients Aij and Bij that minimize anisotropy in the transformed medium. To do so, we create a set of ‘k’ sample points in the non-transformed medium as shown in Fig. 1.

Ideally, the number of points must be greater than the total number of degrees of freedom in the power series, i.e., k>(p+1)(q+1), to avoid oscillations in the polynomial interpolation.

Considering the Cauchy-Riemann condition and the coordinate transformation functions (5) and (6), we calculate the error, E, summed over each sampling point:

E(A,B)=i=0k[(xxyy)2+(xy+yx)2]i
(10)

Taking the gradient of E with respect to the coefficients Aij and Bij, and equating it to zero, one obtains a linear system with 2(p+1)(q+1) equations whose solutions are the coefficients Aij and Bij minimizing E.

Fig. 2 Algorithm used to choose the sampling points for the LSM.
In order to reduce the ratio between peak and mean anisotropy in the transformed medium we selected the sampling points iteratively according to the algorithm presented in Fig. 2.

This algorithm allows a better use of the power series because it reduces the ratio between peak and mean anisotropy to a value less than 'maxR', which must be specified in the project. To this end, the algorithm sample the point with highest anisotropy and excluding (or not) the sampled point with lowest anisotropy. In this procedure, the number of sampled points tends to increase. There are evidences [13

13. D. Liu, L. H. Gabrielli, M. Lipson, and S. G. Johnson, “Transformation inverse design,” Opt. Express 21(12), 14223–14243 (2013). [CrossRef] [PubMed]

] that the peak anisotropy has greater importance than the mean in the device performance.

The ratio reduction between peak and mean anisotropy intensify the reduction of the peak anisotropy for the media and the LSM guarantees that the error function ‘E’, with the sampled points, is the global minima, helping to avoid poor local minima for peak anisotropy that may result of the non-linear optimization. Besides, there are some evidences that the ratio reduction between peak and mean anisotropy, with simultaneous application of the LSM, may be a direction for the global minima of the peak anisotropy.

It is not possible to restrict, algebraically, the refractive index limits, however we can consider different power series orders and chose one of them with lowest anisotropy, satisfying the refractive index limits, as demonstrated in Table 1.

Table 1. Conditions. Peak and Mean Anisotropy. Maximum and Minimum Refractive Index

table-icon
View This Table

The Cauchy-Riemann conditions applied to expressions for the angles θ and φ result in the following expression when anisotropy reduction occurs:

tg(θ)tg(φ)1
(11)

This expression implies that the wave propagation vector in the output tends to be perpendicular to the waveguide output edge when anisotropy reduction occurs, as one would expect. If the medium becomes completely isotropic (which does not occur due to the uniqueness theorem), then we would have equality in expression (11).

It is important to notice that on the boundaries, where both conditions in (7) are simultaneously met, the anisotropy cannot be reduced on the boundary, since it will be solely defined by functions fx and fy, or equivalently, both angles θ and φ can be defined.

In principle, we could extend the application of the LSM for a 3D transformation, achieving conformal map with low anisotropy. However, the 3D transformation is limited, as is demonstrated by the Liouville's theorem for conformal mapping [16

16. C. García-Meca, R. Ortuño, J. Martí, and A. Martínez, “Full three-dimensional isotropic transformation media,” New J. Phys. 16(2), 023030 (2014). [CrossRef]

, 17

17. D. E. Blair, Inversion Theory and Conformal Mapping (American Mathematical Society, 2000).

], by a composition of translations, similarities, orthogonal transformations and inversions, i.e., only the Möbius transformations [4

4. G. E. Shilov, Elementary Real and Complex Analysis (MIT, 1973).

] are conformal in 3D and it could be a startup to application of the LSM for 3D transformations.

3.3 Transformation example

We present a general transformation to create a bent waveguide of arbitrary angle (φ), output position (xo,y0), and expansion or compression factor (e). These parameters are pictured in Fig. 1. We use the general transformation:
x=(xo+ecosφx)(yyf)+x(1yyf)
(12)
y=(y0+esinφx)(yyf)
(13)
Where ‘y = 0’ and ‘yf’ are, respectively, the input and output boundaries, and the original medium is symmetric across the y-axis. This coordinate transformation satisfies the boundary conditions and may become highly anisotropic depending on the chosen parameters.

4. Results

In this section, we present the results of using the LSM for anisotropy reduction in a transformation of the form (12) and (13) with φ=30° and expansion factor e=1.5:

x=(4+1.532x)(y10)+x(1y10)+sin(πy/10)i=0qj=0pAijxiyj
(14)
y=(91.512x)(y10)+sin(πy/10)i=0qj=0pBijxiyj
(15)

The original region is defined in 1.2μmx1.2μm and 0y10μm . The input signal is a Gaussian beam with wavelength 1.55μm applied in the y=0 boundary. The output is measured at y=10μm. In the transformed device we expect a planar Gaussian wave front at the output with 1.5 times wider waist and a 30° rotation in the propagation direction. The first condition in (7) is satisfied allowing to define the angle φ. We considered different polynomial orders for the power series and an evaluation grid of 240 × 1000 points. The parameter maxR used was 3.

After determination of the coefficients with the LSM for each case, the resulting transformation was assessed by their peak and mean anisotropy. Each case and their corresponding results are shown in Table 1. In particular, case 1 represents the original transform with no anisotropy reduction.

The results in Table 1 show the effectiveness of the technique for reducing the peak and mean anisotropy. These results suggest that the residual anisotropy can be made as small as needed (but never exactly zero) by increasing the perturbation polynomial orders. This reduction comes at a price, nonetheless: the increased refractive index contrast, necessary to implement the transformed medium. The waveguide bends designed in [8

8. N. I. Landy and W. J. Padilla, “Guiding light with conformal transformations,” Opt. Express 17(17), 14872–14879 (2009). [CrossRef] [PubMed]

] and [9

9. 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(5), 968–972 (2010). [CrossRef] [PubMed]

], after the reduction of the anisotropy, also resulted in increased refractive index contrast.

Fig. 3 Simple example of coordinate transformation. (a) Non-transformed media. (b) Transformed media with no anisotropy reduction (case 1). (c) Transformed media with anisotropy reduction (case 7).
Figure 3 shows the distortion by the transformation for non-transformed medium, case 1 and case 7. We can see the effect of anisotropy reduction in preserving of the angle between oriented curves.

Fig. 4 Numerical simulations results. (a) Original medium. Only the core and cladding regions are transformed, not the entire simulation domain. (b) Electric filed distribution over the original transformed medium, without anisotropy reduction (case 1). The simulated medium is isotropic, therefore any anisotropy in the transformation results in propagation distortions, as demonstrated by the cross-section plot (line plot) of the device output. (c) Same as (b) but with anisotropy reduction (case 7). The beam propagates through the structure without distortions. (d) Refractive index profile for case 1. (e) Refractive index profile for case 7.
The optimized devices were simulated by finite element method, using the software COMSOL, to evaluate the behavior of the Gaussian beam in the multimode waveguide. The simulated devices were made of isotropic materials, so that the effects of any residual anisotropy would reflect in deformations in the output beam. Figure 4 presents the results for cases 1 and 7 in Table 1.

The results show a significant improvement of the waveguide performance after the reduction of anisotropy. In case 1, the presence of beam deformation (due to mode conversion) is notable, in contrast to case 7, where the beam shape is preserved along the propagation direction (except, of course, for the intended expansion factor of 1.5).

5. Conclusion

This paper presented a new technique for reducing anisotropy in transformation optics with the application of the least squares method. In particular, we developed a mathematical framework to design and to analyze waveguide-based transformations, which was used to exemplify the proposed technique. Our method is able to effectively reduce the anisotropy in the transformed medium close to zero with the benefit of being straightforward to implement.

Acknowledgments

This work is supported by CNPq, CAPES and FAPEMIG

References and links

1.

L. Ahlfors, “On quasi-conformal mappings,” J. Anal. Math. 3(1), 1–58 (1953). [CrossRef]

2.

J. Li and J. B. Pendry, “Hiding under the carpet: A new strategy for cloaking,” Phys. Rev. Lett. 101(20), 203901 (2008). [CrossRef] [PubMed]

3.

U. Leonhardt and T. G. Philbin, “Transformation optics and the geometry of light,” Prog. Opt. 53, 69–152 (2009). [CrossRef]

4.

G. E. Shilov, Elementary Real and Complex Analysis (MIT, 1973).

5.

W. Yan, M. Yan, and M. Qiu, “Necessary and sufficient conditions for reflection less transformation media in an isotropic and homogenous background,” arXiv:0806.3231 (2008).

6.

U. Leonhardt, “Optical conformal mapping,” Science 312(5781), 1777–1780 (2006). [CrossRef] [PubMed]

7.

Z. Chang, X. Zhou, J. Hu, and G. Hu, “Design method for quasi-isotropic transformation materials based on inverse Laplace’s equation with sliding boundaries,” Opt. Express 18(6), 6089–6096 (2010). [CrossRef] [PubMed]

8.

N. I. Landy and W. J. Padilla, “Guiding light with conformal transformations,” Opt. Express 17(17), 14872–14879 (2009). [CrossRef] [PubMed]

9.

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(5), 968–972 (2010). [CrossRef] [PubMed]

10.

N. Kundtz and D. R. Smith, “Extreme-angle broadband metamaterial lens,” Nat. Mater. 9(2), 129–132 (2010). [CrossRef] [PubMed]

11.

H. F. Ma and T. J. Cui, “Three-dimensional broadband and broad-angle transformation-optics lens,” Nat. Commun. 1, 124 (2010).

12.

Q. Wu, Z. H. Jiang, O. Quevedo-Teruel, J. P. Turpin, W. Tang, Y. Hao, and D. H. Werner, “Transformation optics inspired multibeam lens antennas for broadband directive radiation,” IEEE Trans. Antenn. Propag. 61(12), 5910–5922 (2013). [CrossRef]

13.

D. Liu, L. H. Gabrielli, M. Lipson, and S. G. Johnson, “Transformation inverse design,” Opt. Express 21(12), 14223–14243 (2013). [CrossRef] [PubMed]

14.

D. H. Spadoti, L. H. Gabrielli, C. B. Poitras, and M. Lipson, “Focusing light in a curved-space,” Opt. Express 18(3), 3181–3186 (2010). [CrossRef] [PubMed]

15.

K. Astala, T. Iwaniec, and G. Martin, Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane (Princeton University, 2008).

16.

C. García-Meca, R. Ortuño, J. Martí, and A. Martínez, “Full three-dimensional isotropic transformation media,” New J. Phys. 16(2), 023030 (2014). [CrossRef]

17.

D. E. Blair, Inversion Theory and Conformal Mapping (American Mathematical Society, 2000).

OCIS Codes
(000.3860) General : Mathematical methods in physics
(130.2790) Integrated optics : Guided waves
(230.0230) Optical devices : Optical devices
(160.3918) Materials : Metamaterials

ToC Category:
Physical Optics

History
Original Manuscript: May 14, 2014
Revised Manuscript: June 21, 2014
Manuscript Accepted: June 24, 2014
Published: July 23, 2014

Citation
Mateus A. F. C. Junqueira, Lucas H. Gabrielli, and Danilo H. Spadoti, "Anisotropy minimization via least squares method for transformation optics," Opt. Express 22, 18490-18498 (2014)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-15-18490


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References

  1. L. Ahlfors, “On quasi-conformal mappings,” J. Anal. Math.3(1), 1–58 (1953). [CrossRef]
  2. J. Li and J. B. Pendry, “Hiding under the carpet: A new strategy for cloaking,” Phys. Rev. Lett.101(20), 203901 (2008). [CrossRef] [PubMed]
  3. U. Leonhardt and T. G. Philbin, “Transformation optics and the geometry of light,” Prog. Opt.53, 69–152 (2009). [CrossRef]
  4. G. E. Shilov, Elementary Real and Complex Analysis (MIT, 1973).
  5. W. Yan, M. Yan, and M. Qiu, “Necessary and sufficient conditions for reflection less transformation media in an isotropic and homogenous background,” arXiv:0806.3231 (2008).
  6. U. Leonhardt, “Optical conformal mapping,” Science312(5781), 1777–1780 (2006). [CrossRef] [PubMed]
  7. Z. Chang, X. Zhou, J. Hu, and G. Hu, “Design method for quasi-isotropic transformation materials based on inverse Laplace’s equation with sliding boundaries,” Opt. Express18(6), 6089–6096 (2010). [CrossRef] [PubMed]
  8. N. I. Landy and W. J. Padilla, “Guiding light with conformal transformations,” Opt. Express17(17), 14872–14879 (2009). [CrossRef] [PubMed]
  9. Y. G. Ma, N. Wang, and C. K. Ong, “Application of inverse, strict conformal transformation to design waveguide devices,” J. Opt. Soc. Am. A27(5), 968–972 (2010). [CrossRef] [PubMed]
  10. N. Kundtz and D. R. Smith, “Extreme-angle broadband metamaterial lens,” Nat. Mater.9(2), 129–132 (2010). [CrossRef] [PubMed]
  11. H. F. Ma and T. J. Cui, “Three-dimensional broadband and broad-angle transformation-optics lens,” Nat. Commun.1, 124 (2010).
  12. Q. Wu, Z. H. Jiang, O. Quevedo-Teruel, J. P. Turpin, W. Tang, Y. Hao, and D. H. Werner, “Transformation optics inspired multibeam lens antennas for broadband directive radiation,” IEEE Trans. Antenn. Propag.61(12), 5910–5922 (2013). [CrossRef]
  13. D. Liu, L. H. Gabrielli, M. Lipson, and S. G. Johnson, “Transformation inverse design,” Opt. Express21(12), 14223–14243 (2013). [CrossRef] [PubMed]
  14. D. H. Spadoti, L. H. Gabrielli, C. B. Poitras, and M. Lipson, “Focusing light in a curved-space,” Opt. Express18(3), 3181–3186 (2010). [CrossRef] [PubMed]
  15. K. Astala, T. Iwaniec, and G. Martin, Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane (Princeton University, 2008).
  16. C. García-Meca, R. Ortuño, J. Martí, and A. Martínez, “Full three-dimensional isotropic transformation media,” New J. Phys.16(2), 023030 (2014). [CrossRef]
  17. D. E. Blair, Inversion Theory and Conformal Mapping (American Mathematical Society, 2000).

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