## Transformation inverse design |

Optics Express, Vol. 21, Issue 12, pp. 14223-14243 (2013)

http://dx.doi.org/10.1364/OE.21.014223

Acrobat PDF (4343 KB)

### Abstract

We present a new technique for the design of transformation-optics devices based on large-scale optimization to achieve the optimal effective isotropic dielectric materials within prescribed index bounds, which is computationally cheap because transformation optics circumvents the need to solve Maxwell’s equations at each step. We apply this technique to the design of multimode waveguide bends (realized experimentally in a previous paper) and mode squeezers, in which all modes are transported equally without scattering. In addition to the optimization, a key point is the identification of the correct boundary conditions to ensure reflectionless coupling to untransformed regions while allowing maximum flexibility in the optimization. Many previous authors in transformation optics used a certain kind of quasiconformal map which overconstrained the problem by requiring that the entire boundary shape be specified *a priori* while at the same time underconstraining the problem by employing “slipping” boundary conditions that permit unwanted interface reflections.

© 2013 OSA

## 1. Introduction

1. A. J. Ward and J. B. Pendry, “Refraction and geometry in Maxwell’s equations,” J. Mod. Opt. **43**(4):773–793 (1996) [CrossRef] .

8. J. B. Pendry, A. Aubry, D. R. Smith, and S. A. Maier, “Transformation optics and subwavelength control of light,” Science **337**(6094):549–552 (2012) [CrossRef] [PubMed] .

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

21. B. Vasić, G. Isić, R. Gajić, and K. Hingerl, “Coordinate transformation based design of confined metamaterial structures,” Phys. Rev. B **79**(085103) (2009) [CrossRef] .

20. 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**(15):11555–11567 (2008) [CrossRef] [PubMed] .

24. O. Ozgun and M. Kuzuoglu, “Utilization of anisotropic metamaterial layers in waveguide miniaturization and transitions,” IEEE Microw. Wirel. Compon. Lett. **17**(754) (2007) [CrossRef] .

3. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science **312**:1780–1782 (2006) [CrossRef] [PubMed] .

14. T. Han, C. Qiu, J. Dong, X. Tang, and S. Zouhdi, “Homogeneous and isotropic bends to tunnel waves through multiple different/equal waveguides along arbitrary directions,” Opt. Express **19**(14):13020–13030 (2011) [CrossRef] [PubMed] .

22. C. García-Meca, M. M. Tung, J. V. Galán, R. Ortuño, F. J. Rodríguez-Fortuño, J. Martí, and A. Martínez, “Squeezing and expanding light without reflections via transformation optics,” Opt. Express **19**(4):3562–3757 (2011) [CrossRef] [PubMed] .

25. B. Zhang, Y. Luo, X. Liu, and G. Barbastathis, “Macroscopic invisibility cloak for visible light,” Phys. Rev. Lett. **106**(033901) (2011) [CrossRef] .

32. A. V. Novitsky, “Inverse problem in transformation optics,”J. Opt. **13**(035104) (2011) [CrossRef] .

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

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

13. K. Yao and X. Jiang, “Designing feasible optical devices via conformal mapping,” J. Opt. Soc. Am. B **28**(5):1037–1042 (2011) [CrossRef] .

22. C. García-Meca, M. M. Tung, J. V. Galán, R. Ortuño, F. J. Rodríguez-Fortuño, J. Martí, and A. Martínez, “Squeezing and expanding light without reflections via transformation optics,” Opt. Express **19**(4):3562–3757 (2011) [CrossRef] [PubMed] .

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

51. D. R. Smith, Y. Urzhumov, N. B. Kundtz, and N. I. Landy, “Enhancing imaging systems using transformation optics,” Opt. Express **18**(20):21238–21251 (2010) [CrossRef] [PubMed] .

*a priori*specification of the entire boundary shape of the transformation. Further, neither technique can directly incorporate refractive-index bounds. On the other hand, most inverse design in photonics involves repeatedly solving computationally expensive Maxwell equations for different designs [52

52. W. R. Frei, H. T. Johnson, and K. D. Choquette, “Optimization of a single defect photonic crystal laser cavity,” J. Appl. Phys. **103**(033102) (2008) [CrossRef] .

71. Y. Watanabe, N. Ikeda, Y. Sugimoto, Y. Takata, Y. Kitagawa, A. Mizutani, N. Ozaki, and K. Asakawa, “Topology optimization of waveguide bends with wide, flat bandwidth in air–bridge-type photonic crystal slabs,” J. Appl. Phys. **101**(113108) (2007) [CrossRef] .

**x′**(

**x**) that warps light in a desired way (e.g. mapping a straight waveguide to a bend, or mapping an object to a point or the ground for cloaking applications [3

3. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science **312**:1780–1782 (2006) [CrossRef] [PubMed] .

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

40. R. Liu, R. C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science **323**(5912):366–369 (2009) [CrossRef] [PubMed] .

43. T. Ergin, N. Stenger, P. Brenner, J. B. Pendry, and M. Wegener, “Three-dimensional invisibility cloak at optical wavelengths,” Science **328**(5976):337–339 (2010) [CrossRef] [PubMed] .

72. J. H. Lee, J. Blair, V. A. Tamma, Q. Wu, S. J. Rhee, C. J. Summers, and W. Park, “Direct visualization of optical frequency invisibility cloak based on silicon nanorod array,” Opt. Express **17**(15):12922–12928 (2009) [CrossRef] [PubMed] .

73. U. Leonhardt and T. Tyc, “Broadband invisibility by non-euclidian cloaking,” Science **323**(5910):110–112 (2009) [CrossRef] .

*𝒥*=

_{ij}*∂x′*/

_{j}*∂x*to mathematically mimic the effect of the coordinate transformation. This transforms all solutions of Maxwell’s equations in the same way (as opposed to non-TO multimode devices which often have limited bandwidth and/or do not preserve relative phase between modes [60

_{i}60. J. S. Jensen and O. Sigmund, “Systematic design of photonic crystal structures using topology optimization: low-loss waveguide bends,” Appl. Phys. Lett. **84**(12):2022–2024 (2004) [CrossRef] .

71. Y. Watanabe, N. Ikeda, Y. Sugimoto, Y. Takata, Y. Kitagawa, A. Mizutani, N. Ozaki, and K. Asakawa, “Topology optimization of waveguide bends with wide, flat bandwidth in air–bridge-type photonic crystal slabs,” J. Appl. Phys. **101**(113108) (2007) [CrossRef] .

76. V. Liu and S. Fan, “Compact bends for multi-mode photonic crystal waveguides with high transmission and suppressed modal crosstalk,” Opt. Express **21**(7):8069–8075 (2013) [CrossRef] [PubMed] .

84. J. Smajic, C. Hafner, and D. Erni, “Design and optimization of an achromatic photonic crystal bend,” Opt. Express **11**(12):1378–1384 (2003) [CrossRef] [PubMed] .

20. 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**(15):11555–11567 (2008) [CrossRef] [PubMed] .

85. M. Schmiele, V. S. Varma, C. Rockstuhl, and F. Lederer, “Designing optical elements from isotropic materials by using transformation optics,” Phys. Rev. A **81**(033837) (2010) [CrossRef] .

87. M. Rahm, S. A. Cummer, D. Schurig, J. B. Pendry, and D. R. Smith, “Optical design of reflectionless complex media by finite embedded coordinate transformations,” Phys. Rev. Lett **100**(063903) (2008) [CrossRef] [PubMed] .

74. A. Greenleaf, M. Lassas, and G. Uhlmann, “Anisotropic conductivities that cannot be detected by EIT,” Physiol. Meas. **24**(413):413–419 (2003) [CrossRef] [PubMed] .

**x′**(

**x**) yield highly anisotropic and magnetic materials. In principle, these transformed designs can be fabricated with anisotropic microstructures [29

29. 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**(5801):977–980 (2006) [CrossRef] [PubMed] .

88. D. Smith, J. Mock, A. Starr, and D. Schurig, “Gradient index metamaterials,” Phys. Rev. E **71**(036609) (2005) [CrossRef] .

90. F. Xu, R. C. Tyan, P. C. Sun, Y. Fainman, C. C. Cheng, and A. Scherer, “Fabrication, modeling, and characterization of form-birefringent nanostructures,” Opt. Lett. **20**(24):2457–2459 (1995) [CrossRef] [PubMed] .

25. B. Zhang, Y. Luo, X. Liu, and G. Barbastathis, “Macroscopic invisibility cloak for visible light,” Phys. Rev. Lett. **106**(033901) (2011) [CrossRef] .

30. X. Chen, Y. Luo, J. Zhang, K. Jiang, J. B. Pendry, and S. Zhang, “Macroscopic invisibility cloaking of visible light,” Nat. Commun. **2**(176) (2011) [CrossRef] .

*n*

_{min}and

*n*

_{max}of the fabrication process (for example, subwavelength nanostructures [36

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

41. J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. **8**:568–571 (2009) [CrossRef] [PubMed] .

43. T. Ergin, N. Stenger, P. Brenner, J. B. Pendry, and M. Wegener, “Three-dimensional invisibility cloak at optical wavelengths,” Science **328**(5976):337–339 (2010) [CrossRef] [PubMed] .

47. L. H. Gabrielli and M. Lipson, “Transformation optics on a silicon platform,” J. Opt. **13**(024010) (2011) [CrossRef] .

72. J. H. Lee, J. Blair, V. A. Tamma, Q. Wu, S. J. Rhee, C. J. Summers, and W. Park, “Direct visualization of optical frequency invisibility cloak based on silicon nanorod array,” Opt. Express **17**(15):12922–12928 (2009) [CrossRef] [PubMed] .

88. D. Smith, J. Mock, A. Starr, and D. Schurig, “Gradient index metamaterials,” Phys. Rev. E **71**(036609) (2005) [CrossRef] .

91. H. Kurt and D. S. Citrin, “Graded index photonic crystals,” Opt. Express **15**(3):1240–1253 (2007) [CrossRef] [PubMed] .

93. U. Levy, M. Abashin, K. Ikeda, A. Krishnamoorthy, J. Cunningham, and Y. Fainman, “Inhomogenous dielectric metamaterials with space-variant polarizability,” Phys. Rev. Lett. **98**(243901) (2007) [CrossRef] .

94. L. Gabrielli and M. Lipson, “Integrated luneburg lens via ultra-strong index gradient on silicon,” Opt. Express **19**(21):20122–20127 (2011) [CrossRef] [PubMed] .

99. J. Brazas, G. Kohnke, and J. McMullen, “Mode-index waveguide lens with novel gradient boundaries developed for application to optical recording,” Appl. Opt. **31**(18):3420–3428 (1992) [CrossRef] [PubMed] .

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

*quasiconformal maps*[which in mathematical analysis are defined as

*any*orientation-preserving transformation with bounded anisotropy (as quantified in Sec. 2.4)]. However, in transformation optics the term “quasiconformal” has become confusingly associated with only a single choice of quasiconformal map suggested by Li and Pendry [33

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

*constant stretching*(and thus anisotropy) everywhere. This map, which also happens to minimize the peak anisotropy given the slipping boundary conditions [33

**101**(203901) (2008) [CrossRef] .

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

37. N. I. Landy, N. Kundtz, and D. R. Smith, “Designing three-dimensional transformation optical media using quasiconformal coordinate transformations,” Phys. Rev. Lett. **105**(193902) (2010) [CrossRef] .

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

41. J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. **8**:568–571 (2009) [CrossRef] [PubMed] .

43. T. Ergin, N. Stenger, P. Brenner, J. B. Pendry, and M. Wegener, “Three-dimensional invisibility cloak at optical wavelengths,” Science **328**(5976):337–339 (2010) [CrossRef] [PubMed] .

**x**

*′*at the interface [87

87. M. Rahm, S. A. Cummer, D. Schurig, J. B. Pendry, and D. R. Smith, “Optical design of reflectionless complex media by finite embedded coordinate transformations,” Phys. Rev. Lett **100**(063903) (2008) [CrossRef] [PubMed] .

105. W. Yan, M. Yan, Z. Ruan, and M. Qiu, “Coordinate transformations make perfect invisibility cloaks with arbitrary shape,” New J. Phys. **10**(043040) (2008) [CrossRef] .

*𝒥*as well. If one fixes the transformation on part or all of the boundary (instead of just the corners) and minimizes the peak anisotropy, the result is called (in analysis) an

*extremal*quasiconformal map [106–111]. We point out in Sec. 2.2 that this extremal quasiconformal map can never be conformal except in trivial cases. Additionally, previous work in quasiconformal transformation optics underconstrained the space of transformations in one way but overconstrained it in another. Li and Pendry’s method, along with other work on extremal quasiconformal maps in mathematical analysis, assumed that the entire boundary

*shape*of the transformed domain is specified

*a priori*(even if the

*value*of the transformation at the boundary is not specified). In contrast, transformation inverse design allows parts of the boundary shape to be freely chosen by the optimization, only fixing aspects of the boundary that are determined by the underlying problem (e.g. the input/output facets of the boundary in Fig. 1) as explained in Secs. 3.2, allowing a much larger space of transformations to be searched. Also, for such stricter boundary conditions, minimizing the mean anisotropy is

*not*equivalent to minimizing the peak anisotropy [107, 112

112. Z. Balogh, K. Fässler, and I. Platis, “Modulus of curve families and extremality of spiral-stretch maps,” J. Anal. Math. **113**(1):265–291 (2011) [CrossRef] .

114. Z. Balogh, K. Fässler, and I. Platis, “Modulus method and radial stretch map in the Heisenberg group,” Ann. Acad. Sci. Fenn. **38**(1):149–180 (2013) [CrossRef] .

117. L. Gabrielli, D. Liu, S. G. Johnson, and M. Lipson, “On-chip transformation optics for multimode waveguide bends,” Nat. Commun. **3**(1217) (2012) [CrossRef] [PubMed] .

## 2. Mathematical preliminaries

### 2.1. Transformation optics

*e*

^{−iωt}), without sources or currents, in linear isotropic dielectric media [

**=**

*ε**ε*(

**x**),

**=**

*μ**μ*

_{0}] are Consider a coordinate transformation

**x′**(

**x**) with Jacobian

*primed*gradient vector as

**E′**≡

*𝒥*

^{−1}

**E**and

**H′**≡

*𝒥*

^{−1}

**H**. One can then rewrite Eq. (1), after some rearrangement [1

1. A. J. Ward and J. B. Pendry, “Refraction and geometry in Maxwell’s equations,” J. Mod. Opt. **43**(4):773–793 (1996) [CrossRef] .

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

*transformation optics*(TO). It is actually the specific case of a much more general result from general relativity [120

120. J. Plebanski, “Electromagnetic waves in gravitational fields,” Phys. Rev. **118**(5):1396–1408 (1960) [CrossRef] .

121. U. Leonhardt and T. G. Philbin, “General relativity in electrical engineering,” New J. Phys. **8**(10) (2006) [CrossRef] .

*continuous*transformation of all space that is the identity

**x′**(

**x**) =

**x**in the “untransformed” regions, as depicted in Fig. 2. More generally, the untransformed regions can be simple rotations or translations, but when examining a particular interface, we can always choose the coordinates to be

**x′**=

**x**at that interface. It is clear by construction that continuous

**x′**is sufficient for reflectionless interfaces [87

87. M. Rahm, S. A. Cummer, D. Schurig, J. B. Pendry, and D. R. Smith, “Optical design of reflectionless complex media by finite embedded coordinate transformations,” Phys. Rev. Lett **100**(063903) (2008) [CrossRef] [PubMed] .

104. L. Bergamin, “Electromagnetic fields and boundary conditions at the interface of generalized transformation media,” Phys. Rev. A **80**(063835) (2009) [CrossRef] .

105. W. Yan, M. Yan, Z. Ruan, and M. Qiu, “Coordinate transformations make perfect invisibility cloaks with arbitrary shape,” New J. Phys. **10**(043040) (2008) [CrossRef] .

**x′**(

**x**) continuous at the interface, we show below that an

*isotropic*transformation will also have a continuous

*𝒥*at the interface. These boundary conditions are essential for designing useful transformations without interface reflections.

### 2.2. Transformations to isotropic dielectric materials

**x′**(

**x**) is 2D (

*z′*=

*z*and

*𝒥*block-diagonal (with the

*zz*element independent of the

*xy*block). Then, the

*xy*block of

*𝒥*is isotropic if and only if the diagonal elements are equal and the off-diagonal elements vanish: In this case, the

^{T}𝒥*𝒥*part of Eq. (3) becomes

**E**=

*Eẑ*and

**H**

*· ẑ*= 0) versus transverse-electric (TE) polarized modes (which have

**E**

*ẑ*= 0 and

**H**=

*Hẑ*). For TM-polarized modes, the fields

**E′**,

**H′**in the primed coordinate system are also TM-polarized and Eq. (2) becomes Hence, for TM modes, an isotropic transformation can be exactly mapped to an isotropic dielectric material. Similarly, for TE-polarized modes the equivalent material is isotropic and

*magnetic*. However, if det

*𝒥*varies slowly compared to the wavelengths of the fields, then the transformation can

*approximately*still be mapped to an isotropic dielectric material by making an eikonal approximation (as in [119] Ch. 8.10) and commuting 1/

*μ′*with one of the curls in the Maxwell equations. In particular, Eq. (2) can be written: where the last term can be neglected for slowly varying transformations. Because the TM case is conceptually simpler and does not require this extra approximation, we work with it exclusively for the rest of this paper. Also, because the non-trivial aspects of the transformation occur in the

*xy*plane, we hereafter use

*𝒥*to denote the

*xy*block.

### 2.3. Conformal maps and uniqueness

*𝒥*, then the transformation preserves angles in the

*xy*plane. Additionally, if det

*𝒥*> 0, then the transformation also preserves handness and orientation. The combination of these two properties is called a conformal map [100], and is the only case where the situation in Sec. 2.2 can be realized. We only consider transformations with det

*𝒥*> 0 in order to restrict ourselves to dielectric materials. Also, a det

*𝒥*> 0 transformation coupled continuously to an untransformed (det

*𝒥*= 1) region would require singularities (det

*𝒥*= 0) at some points. Conformal maps are described by analytic functions, which are of the form

*x′*+

*iy′*=

*w′*(

*w*) (where

*w*≡

*x*+

*iy*is the untransformed complex coordinate) and whose real and imaginary parts satisfy the Cauchy–Riemann equations of complex analysis [100, 101].

*w′*(

*w*) =

*w*in some region, then

*w′*(

*w*) =

*w*everywhere (similarly for a simple rotation or translation in some regions).

*𝒥*, not just a continuous

**x′**(

**x**). It is easy to see this explicitly in the example of Fig. 2: continuity of

**x′**(

**x**) at the interface requires that

*𝒥*. The isotropy of

*𝒥*then forces

^{T}𝒥*𝒥*= 𝕀. Therefore, in the sections that follow (where we search for

*approximately*isotropic maps), we will impose the condition of continuous

*𝒥*as a boundary condition on our transformations. The resulting transformations are nearly isotropic in the interior and exactly isotropic on the interfaces. This condition, discussed at the end of Sec. 2.5, also has the useful consequence of producing a continuous refractive index

### 2.4. Quasiconformal maps and measures of anisotropy

*approximated*by an isotropic material at the cost of some scattering corrections to the exactly transformed modes of the nearly isotropic material. To do this, one must first quantify the measure of anisotropy that is to be minimized. The isotropy condition of Eq. (4) is equivalent to

*λ*

_{1}=

*λ*

_{2}, where

*λ*

_{1}(

*x*,

*y*) ≥

*λ*

_{2}(

*x*,

*y*) are the two eigenvalues of

*𝒥*. While

^{T}𝒥*λ*

_{1}−

*λ*

_{2}works as a measure of anisotropy, it is convenient for optimization purposes to define

*differentiable*measures that can be expressed directly in terms of the trace and determinant of

*𝒥*, and precisely such quantities have been developed in the literature on quasiconformal maps [106, 107, 111, 122

122. L. Ahlfors, “On quasiconformal mappings. J. Anal. Math. **3**(1):1–58 (1953) [CrossRef] .

123. W. Zeng, F. Luo, S. T. Yau, and X. D. Gu, “Surface quasi-conformal mapping by solving Beltrami equations,” Proc. Math. Surfaces **XIII**391–408 (2009) [CrossRef] .

### 2.5. Scalarization errors for nearly isotropic materials

*scalarized*(as in [33

**101**(203901) (2008) [CrossRef] .

**E**

_{0},

**H**

_{0}can be mapped to a transformed material and geometry that is also isotropic dielectric and guides TM modes

**E′**

_{0},

**H′**

_{0}. This is exact for 𝕂 = 1, but for a

*nearly*isotropic transformation with 𝕂 > 1, the equivalent permeability is

**= 𝕀 + Δ**

*μ′***, where the anisotropic part Δ**

*μ***is proportional to 𝕂 − 1 to lowest order. While Δ**

*μ***≠ 0 cannot be fabricated using dielectric gradient index processes, one can neglect this small correction so that the actual fabricated material has permeability**

*μ′*

*μ′*_{approx}= 𝕀. In practice, we absorb any Δ

**into**

*μ′**ε′*by

*ε′*multiplying by the average eigenvalue of

**but this does not change the**

*μ′**𝒪*(𝕂 − 1) error.

126. S. G. Johnson, M. L. Povinelli, M. Soljačić, A. Karalis, S. Jacobs, and J. D. Joannopoulos, “Roughness losses and volume-current methods in photonic-crystal waveguides,” Appl. Phys. B **81**(283–293) (2005) [CrossRef] .

**and Δ**

*ε***will generically lead to scattered fields with magnitudes of**

*μ**𝒪*(|Δ

**|+|Δ**

*ε***|) and scattered power of**

*μ**𝒪*(|Δ

**|**

*ε*^{2}+|Δ

**|**

*μ*^{2}). The modes of the approximate scalarized material

*μ′*_{approx},

*ε′*

_{approx}are then the exact guided modes plus scattered power corrections of

*𝒪*(|Δ

**|**

*μ*^{2}) =

*𝒪*[(𝕂 − 1)

^{2}].

*𝒥*at the input/output facets of the domain. As explained in Sec. 2.3, a purely isotropic transformation in the neighborhood of the interface, along with a continuity of

**x′**, would automatically yield continuous

*𝒥*, so one might hope that minimizing anisotropy would suffice to obtain a nearly continuous

*𝒥*. Unfortunately, as we show in the Appendix, the resulting discontinuity in det

*𝒥*(and hence the discontinuity in the refractive index) is of order

*𝒪*(𝕂 − 1) power loss due to reflections, much larger than the

*𝒪*[(𝕂 − 1)

^{2}] power scattering from anisotropy in the interior. This would make it pointless to minimize the anisotropy in the interior, since the boundary reflections would dominate. In fact, our initial implementation of the bend optimization in Sec. 3.4 did not enforce continuity of

*𝒥*, and we obtained a large 2% index discontinuity at the endfacets for max

**𝕂 − 1 ≈ 0.0005. Therefore, in Sec. 3.4 we impose continuity of**

_{x}*𝒥*explicitly.

### 2.6. General optimization of anisotropy

**x′**and its Jacobian

*𝒥*, as well as the engineering fabrication bounds

*n*

_{min}and

*n*

_{max}. By using numerical optimization, we can in principle achieve both a lower mean anisotropy and a lower peak anisotropy than by traditional quasiconformal mapping, since the optimization is also free to vary the boundary shape (with at most the input/output interfaces fixed, although in some cases their locations and shapes are allowed to vary as well). The minimization problem can be written, for example, as where 〈𝕂(

**x**)〉 is a

*functional*norm taken over the domain of

**x′**(

**x**). We consider two possible norms: the

*L*

_{1}norm (the mean 〈𝕂〉

**), and the**

_{x}*L*

_{∞}norm (max

**𝕂). We show in Sec. 4.2 that minimizing the mean can lead to pockets of high anisotropy which can cause increased scattering. Directly optimizing the peak anisotropy on the other hand, avoids such pockets while simultaneously keeping the mean nearly as low. The continuity of**

_{x}**x′**and

*𝒥*at the input/output interfaces, as well as other constraints on the interface locations, are imposed implicitly by the parametrization of

**x′**(

**x**) (as explained in Sec. 3.4).

## 3. Multimode Bend design

14. T. Han, C. Qiu, J. Dong, X. Tang, and S. Zouhdi, “Homogeneous and isotropic bends to tunnel waves through multiple different/equal waveguides along arbitrary directions,” Opt. Express **19**(14):13020–13030 (2011) [CrossRef] [PubMed] .

18. Z. L. Mei and T. J. Cui, “Experimental realization of a broadband bend structure using gradient index metamaterials,” Opt. Express **17**(20):18354–18363 (2009) [CrossRef] [PubMed] .

20. 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**(15):11555–11567 (2008) [CrossRef] [PubMed] .

21. B. Vasić, G. Isić, R. Gajić, and K. Hingerl, “Coordinate transformation based design of confined metamaterial structures,” Phys. Rev. B **79**(085103) (2009) [CrossRef] .

**17**(17):14872–14879 (2009) [CrossRef] [PubMed] .

13. K. Yao and X. Jiang, “Designing feasible optical devices via conformal mapping,” J. Opt. Soc. Am. B **28**(5):1037–1042 (2011) [CrossRef] .

### 3.1. Simple circular bends

*L*and width unity (in arbitrary distance units to be determined later) into a bend with inner radius

*R*and outer radius

*R*+ 1 (as shown in Fig. 3). For convenience, we choose the untransformed coordinates to be

*R*≤

*x*≤

*R*+1 and

**x′**(

**x**) can be written as where

*r*=

*x*and

**x′**is continuous at the input/output interfaces

*𝒥*is not continuous there, which can be seen from

**≠**

*μ′**μ*

_{0}𝕀 is highly anisotropic. The anisotropy for this transformation is

*R*≫ 1 at the outer radius

*x*=

*R*+1. Note that one can instead choose

*conformal*bend

**x′**nor

*𝒥*are continuous at the input/output interfaces, leading to large reflections there.

### 3.2. Generalized bend transformations

*and*continuous-interface transformations of the form of Eq. (14), where the intermediate polar coordinates are now arbitrary functions

*r*(

*x*,

*y*) and

*θ*(

*x*,

*y*). The ratio

*L/R*is now an optimization parameter. The Jacobian then satisfies We find that the optimization always seems to prefer a symmetric bend (and if the optimum is unique, it must be symmetric), so we impose a mirror symmetry in order to halve our search space: We also require interface continuity of

**x′**and

*𝒥*(as discussed in Sec. 2.3), which give the conditions at

### 3.3. Numerical optimization problem

*n′*to lie between some values

*n*

_{min}and

*n*

_{max}. We choose units so that the width of the transformed region is unity (

*R*≤

*x*≤

*R*+ 1), and consider transforming a straight waveguide of width Δ

_{w}< 1. Δ

_{w}should be small enough so that the exponential tails of the waveguide modes are negligible outside the transformed region. In the straight waveguide segment to be transformed (as well as the straight waveguides to be coupled into the input and output interfaces of the bend),

*n*(

**x**) is high in the core

*n*(

**x**) =

*n*

_{0}

*p*(

*x*) of an overall refractive index

*n*

_{0}and a normalized profile

*p*(

*x*) that is unity in the cladding and some value greater than unity in the core (determined by the ratio of the high and low index regions of the straight waveguide). The transformed refractive index is given by where the average eigenvalue

*μ′*of the magnetic permeability Eq. (12) has been absorbed into the dielectric index. The overall refractive-index scaling

*n*

_{0}is then allowed to freely vary as a parameter in the optimization. Second, like the circular TO bend, the optimum TO bend is expected to have a tradeoff between the bend radius and anisotropy. Because of this expected tradeoff, we can choose to either minimize

*R*while keeping 𝕂 fixed, or minimize 𝕂 while keeping

*R*fixed. We focus on the latter choice, since the bend radius is the more intuitive target quantity to know beforehand. Also, we find empirically that optimizing 𝕂 converges much faster than optimizing

*R*while yielding the same local minima.

**𝕂 with**

_{x}**x**∈

*G*for some grid

*G*of some points to be defined in Sec. 3.4. However, the peak (the

*L*

_{∞}norm) is not a differentiable function of the design parameters, so it should not be directly used as the objective function. Instead, we perform a standard transformation [116]: we introduce a dummy variable

*t*and indirectly minimize the peak 𝕂 using a differentiable inequality constraint between

*t*and 𝕂(

**x**) at all

**x**∈

*G*:

*L*

_{∞}norm is better to minimize than the

*L*

_{1}norm (the mean anisotropy).

*L*

_{1}norm, 〈𝕂〉

**= ∫ 𝕂d**

_{x}*x*d

*y*/area [which

*is*differentiable in terms of the parameters

*r*(

**x**),

*θ*(

**x**),

*n*

_{0}, and

*L*] is implemented as

*r*=

*x*,

*r*(

**x**),

*θ*(

**x**) by perturbing from this base case. (We only perform

*local*optimization; not global optimization, but comment in Sec. 4.3 on a simple technique to avoid being trapped in poor local minima.) As explained in Sec. 3.4, the perturbations will be parameterized such that the symmetry and continuity constraints are satisfied automatically. Figure 3 shows a schematic of the bend transformation optimization process. First, the straight region is mapped to a circular bend. Then, the intermediate polar coordinates

*r*and

*θ*for every point

**x**are perturbed, using an optimization algorithm described at the end of Sec. 3.4, and the desired norm (either

*L*

_{1}and

*L*

_{∞}) of the anisotropy is computed. This process is repeated at each optimization step until the structure converges to a local minimum in 〈𝕂〉.

### 3.4. Spectral parameterization

*r*and

*θ*can be written as the circular bend transformation plus perturbations parametrized in the spectral basis [115, 116]: where the coordinate 2

*x*− 2

*R*− 1 has been centered appropriately for the domain [−1, 1] of degree-

*ℓ*Chebyshev polynomials

*T*. The sines and cosines have been chosen to satisfy the mirror-symmetry conditions of Eq. (17). The sine series also automatically satisfies the second continuity condition of Eq. (18). In order to satisfy the rest of the conditions, the following constraints are also imposed: These equations are solved to simply eliminate the

_{ℓ}*N*×

_{ℓ}*N*< 100) of spectral coefficients

_{m}*C*are needed to achieve very low-anisotropy (𝕂−1 ≈ 10

^{r,}^{θ}^{−4}) transformations. Second, if the fabrication process favors slowly varying transformations (or if these are needed to make the eikonal approximation for the TE polarization, as in Sec. 2.2), this constraint may be imposed simply by using smaller

*N*and

_{ℓ}*N*.

_{m}129. M. J. D. Powell, “A direct search optimization method that models the objective and constraint functions by linear interpolation,” Adv. Optim. Numer. Anal. (1994) [CrossRef] .

130. M. J. D. Powell, “Direct search algorithms for optimization calculations,” Acta Numer. **7**:287–336 (1998) [CrossRef] .

131. S. G. Johnson, The NLopt nonlinear-optimization package (http://ab-initio.mit.edu/nlopt) (2007).

*𝒥*and det

^{T}𝒥*𝒥*, which determine all the non-trivial objective and constraint functions in this optimization problem, are so computationally inexpensive to evaluate that the convergence rate is not a practical concern.

## 4. Optimization results

### 4.1. Minimal peak anisotropy

_{∞}design is shown in Fig. 4, along with the scalarized circular TO bend for comparison. The bend radius was

*R*= 2 and the number of spectral coefficients was

*N*= 5,

_{ℓ}*N*= 8. The objective and constraints were evaluated on a 100 × 140 grid

_{m}*G*in

**x**(Chebyshev points in the

*x*direction and a uniform grid in the

*y*direction). This design had max

**𝕂 − 1 ≈ 5 × 10**

_{x}^{−4}and mean 〈𝕂〉 − 1 ≈ 10

^{−4}. In comparison, the circular TO bend of the same radius has max

**𝕂 − 1 ≈ 0.1 and 〈𝕂〉 − 1 ≈ 10**

_{x}^{−2}.

*R*= 2 optimized design structure was compared in finite-element Maxwell simulations (using the FEniCS code [132

132. A. Logg, K. A. Mardal, and G. N. Wells, *Automated Solution of Differential Equations by the Finite Element Method* (Springer, 2012) [CrossRef] .

*n′*(

**x′**) =

*n*(

**x**)] and the scalarized circular TO bend. The four lowest-frequency modes of a multimode straight waveguide were injected at the input interface

*T*was computed using the measured fields at the output interface

*θ̂*is the propagation direction of the guided modes,

*j*th exactly guided mode of the non-scalarized material (

**,**

*μ′**ε′*), and

**H**

*is the actual magnetic field of the approximate scalarized material at the interface after injecting a normalized mode*

_{i}*T*equal to the power scattered into the

_{ij}*j*th output mode from the

*i*th input mode. For a straight waveguide, which has no intermodal scattering,

*T*= 𝕀. Figure 4 shows a dramatically improved

*T*for the scalarized and optimized TO bend compared to the scalarized circular TO bend. [The rows and columns of

*T*for the circular bend add up to less than one because some power has either been scattered out of the waveguide entirely, or some power has been scattered into fifth or higher-order modes. The rows and columns of

*T*for the optimized bend add up to nearly 1, with the small deficiency due to the

*𝒪*(𝕂 − 1) out-of-bend and higher-order intermodal scattering as well as mesh-descretization error.]

### 4.2. Minimizing max versus minimizing mean

*R*= 2.5,

*N*= 3, and

_{ℓ}*N*= 6 are shown in Fig. 6. Both structures had very low mean anisotropy 〈𝕂〉

_{m}**− 1. The mean-minimized structure, at 〈𝕂〉 − 1 ≈ 10**

_{x}^{−5}, had a slightly lower mean than the peak-minimized structure which had 〈𝕂〉 − 1 ≈ 1.5 × 10

^{−5}. However, in terms of the peak anisotropy, the peak-minimized structure is the clear winner by a factor of 2.5, with max

**𝕂 − 1 ≈ 2 × 10**

_{x}^{−4}as opposed to max

**𝕂 − 1 ≈ 5 × 10**

_{x}^{−4}for the mean-optimized structure. Both structures were scalarized and tested in finite-element Maxwell simulations of the four lowest-frequency modes of the straight waveguide. The scattered-power matrix shows that the difference in max

**𝕂 resulted in an order of magnitude reduction in the intermodal scattering (as shown in the off-diagonal elements) and noticeably improved transmission, (especially in the element**

_{x}*T*

_{44}= 0.89 for the fourth mode).

### 4.3. Tradeoff between anisotropy and radius

**𝕂 for the optimized bend, similar to the circular TO bend, decreases monotonically with**

_{x}*R*(as shown in Fig. 7). Unlike the circular bend, however, this tradeoff seems asymptotically

*exponential*rather than

*𝒪*(

*R*

^{−2}). In particular, there are two clearly different regimes for this tradeoff: a power law 𝕂 − 1 ∼

*R*

^{−4}at small

*R*≲ 3 and an exponential decay 𝕂 − 1 ∼ exp(−0.34

*R*), at larger

*R*. The second regime was only attained after using successive optimization, because with only one independent optimization run the algorithm tended to get stuck in local minima. For successive optimization, the optimum structure is used as a starting guess for the next run, and the initial step size is set large enough so that the algorithm can reach better local minima than the previous one.

*R*≲ 3, we found that there are multiple local minima and that independent optimizations for different

*R*tend to be trapped in suboptimal local minima, as shown by the open dots in Fig. 7. To avoid this problem, we used a “successive optimization” technique in which the optimal structure for smaller

*R*is rescaled as the starting guess for local optima at a larger

*R*, in order to stay along the exponential-tradeoff curve. (Another possible heuristic is “successive refinement” [133

133. A. Oskooi, A. Mutapcic, S. Noda, J. D. Joannopoulos, S. P. Boyd, and S. G. Johnson, “Robust optimization of adiabatic tapers for coupling to slow-light photonic-crystal waveguides,” Opt. Express **20**(19):21558–21575 (2012) [CrossRef] [PubMed] .

*N*are used as starting points for optimizing using larger

_{ℓ,m}*N*)

_{ℓ,m}### 5. Mode squeezer

22. C. García-Meca, M. M. Tung, J. V. Galán, R. Ortuño, F. J. Rodríguez-Fortuño, J. Martí, and A. Martínez, “Squeezing and expanding light without reflections via transformation optics,” Opt. Express **19**(4):3562–3757 (2011) [CrossRef] [PubMed] .

*x*≤ 1 and 0 ≤

*y*≤

*L*. The goal of this transformation

**x′**(

**x**) is to focus the beam by minimizing the

*mid*-

*beam width*As in Sec. 3.4, the transformation is written as a perturbation from the identity transformation (which was used as the starting guess) and parameterized in the spectral basis The sine series automatically satisfies mirror symmetry about

**x′**at the input/output interfaces

*y*= 0,

*L*. However, we found that constraining the coefficients

*C*to enforce continuity of

^{x,y}*𝒥*(as in Sec. 3.4) was not necessary (although it might give a better result) since the optimization algorithm only squeezed the center region while leaving the interfaces and the regions around them relatively untouched. In this problem, we could either minimize 𝕂 for a fixed

*W*or minimize

*W*for a fixed 𝕂, and we happened to choose the latter.

*β*> 0 and 0 <

*α*< 1. Superficially, the design seems similar to an “adiabatic” taper between a wide low-index waveguide and a narrow high-index waveguide, and it is known that any sufficiently gradual taper of this form would have low scattering due to the adiabatic theorem [137

137. S. G. Johnson, P. Bienstman, M. A. Skorobogatiy, M. Ibanescu, E. Lidorikis, and J. D. Joannopoulos, “Adiabatic theorem and continuous coupled-mode theory for efficient taper transitions in photonic crystals,” Phys. Rev. E **66**(066608) (2002) [CrossRef] .

*increases*the scattering.

## 6. Concluding remarks

*𝒥*at the input/output interfaces, there is no conceptual reason why those interfaces need be flat. A better bend, for example, might be designed by constraining the location of only two corners (to fix the bend radius) and constraining only

*𝒥*on other parts of the endfacets. However, we already achieve an exponential tradeoff between radius and anisotropy, so we suspect that further relaxing the constraints would only gain a small constant factor rather than yielding an asymptotically faster tradeoff. In the case of the mode squeezer, one could certainly achieve better results by imposing the proper

*𝒥*constraints at the endfacets. It would also be interesting to apply similar techniques to ground-plane cloaking.

*wants*to discriminate between modes (e.g. a modal filter) or to scatter light between modes (e.g. a mode transformer). TO is ideal for devices in which it is desirable that all modes be transported equally, with no scattering. Even for the latter case (such as our multimode bend), however, TO designs almost certainly trade off computational convenience for optimality, because they impose a stronger constraint than is strictly required: TO is restricted to designs where the solutions at

*all points*in the design are coordinate transformations of the original system, whereas most devices are only concerned with the solutions at the endfacets. For example, it is conceivable that a more compact multimode bend could be designed by allowing intermodal scattering

*within*the bend as long as the modes scatter back to their original configurations by the endfacet; the interior of the bend might not even be a waveguide, and instead might be a resonant cavity of some sort [58

58. L. Frandsen, A. Harpøth, P. Borel, M. Kristensen, J. Jensen, and O. Sigmund, “Broadband photonic crystal waveguide 60° bend obtained utilizing topology optimization,” Opt. Express **12**(24):5916–5921 (2004) [CrossRef] [PubMed] .

77. A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. **77**:3787–3790 (1996) [CrossRef] [PubMed] .

78. C. Ma, Q. Zhang, and E. V. Keuren, “Right-angle slot waveguide bends with high bending efficiency,” Opt. Express **16**(19):14330 (2008) [CrossRef] [PubMed] .

138. C. Manolatou, S. G. Johnson, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “High-density integrated optics,” J. Lightwave Technol. **17**(9):1682–1692 (1999) [CrossRef] .

## Appendix

*𝒥*. In particular, we examine the Jacobian

*𝒥*for nearly isotropic transformations (𝕂 ≈ 1) that also have

**x′**=

**x**explicitly constrained at the interfaces. (The following analysis can also be straight-fowardly extended to situations where

**x′**is a simple rotation of

**x**on the interface, or where the interface has an arbitrary shape.) In this case, the Jacobian is where

*𝒥*is nearly isotropic. The anisotropy Eq. (11) is then: The determinant then satisfies This square-root dependence is also reflected in the refractive index

^{T}𝒥*𝒪*(𝕂 − 1) power loss due to interface reflections that overwhelm the

*𝒪*[(𝕂 − 1)

^{2}] corrections to scattered power due to the scalarization of nearly isotropic transformations (as explained in Sec. 2.5). Hence, it becomes necessary to explicitly constrain

*𝒥*= 𝕀

*in addition*to

**x′**=

**x**.

## Acknowledgment

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76. | V. Liu and S. Fan, “Compact bends for multi-mode photonic crystal waveguides with high transmission and suppressed modal crosstalk,” Opt. Express |

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78. | C. Ma, Q. Zhang, and E. V. Keuren, “Right-angle slot waveguide bends with high bending efficiency,” Opt. Express |

79. | V. Liu and S. Fan, “Compact bends for multi-mode photonic crystal waveguides with high transmission and suppressed modal crosstalk,” Opt. Express |

80. | A. Chutinan, M. Okano, and S. Noda, “Wider bandwidth with high transmission through waveguide bends in two-dimensional photonic crystal slabs,” Appl. Phys. Lett. |

81. | A. Chutinan and S. Noda, “Highly confined waveguides and waveguide bends in three-dimensional photonic crystal,” Appl. Phys. Lett. |

82. | B. Chen, T. Tang, and H. Chen, “Study on a compact flexible photonic crystal waveguide and its bends,” Opt. Express |

83. | Y. Zhang and B. Li, “Photonic crystal-based bending waveguides for optical interconnections,” Opt. Express |

84. | J. Smajic, C. Hafner, and D. Erni, “Design and optimization of an achromatic photonic crystal bend,” Opt. Express |

85. | M. Schmiele, V. S. Varma, C. Rockstuhl, and F. Lederer, “Designing optical elements from isotropic materials by using transformation optics,” Phys. Rev. A |

86. | D. Schurig, J. B. Pendry, and D. R. Smith, “Transformation-designed optical elements,” Opt. Express |

87. | M. Rahm, S. A. Cummer, D. Schurig, J. B. Pendry, and D. R. Smith, “Optical design of reflectionless complex media by finite embedded coordinate transformations,” Phys. Rev. Lett |

88. | D. Smith, J. Mock, A. Starr, and D. Schurig, “Gradient index metamaterials,” Phys. Rev. E |

89. | D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science |

90. | F. Xu, R. C. Tyan, P. C. Sun, Y. Fainman, C. C. Cheng, and A. Scherer, “Fabrication, modeling, and characterization of form-birefringent nanostructures,” Opt. Lett. |

91. | H. Kurt and D. S. Citrin, “Graded index photonic crystals,” Opt. Express |

92. | B. Vasić, R. Gajić, and K. Hingerl, “Graded photonic crystals for implementation of gradient refractive index media,” J. Nanophotonics |

93. | U. Levy, M. Abashin, K. Ikeda, A. Krishnamoorthy, J. Cunningham, and Y. Fainman, “Inhomogenous dielectric metamaterials with space-variant polarizability,” Phys. Rev. Lett. |

94. | L. Gabrielli and M. Lipson, “Integrated luneburg lens via ultra-strong index gradient on silicon,” Opt. Express |

95. | Y. Wang, C. Sheng, H. Liu, Y.J. Zheng, C. Zhu, S. M. Wang, and S. N. Zhu, “Transformation bending device emulated by graded-index waveguide,” Opt. Express |

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117. | L. Gabrielli, D. Liu, S. G. Johnson, and M. Lipson, “On-chip transformation optics for multimode waveguide bends,” Nat. Commun. |

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124. | J. F. Thompson, B. K. Soni, and N. P. Weatherill, |

125. | P. Knupp and S. Steinberg, |

126. | S. G. Johnson, M. L. Povinelli, M. Soljačić, A. Karalis, S. Jacobs, and J. D. Joannopoulos, “Roughness losses and volume-current methods in photonic-crystal waveguides,” Appl. Phys. B |

127. | A. W. Snyder and J. D. Love, |

128. | W. C. Chew, |

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132. | A. Logg, K. A. Mardal, and G. N. Wells, |

133. | A. Oskooi, A. Mutapcic, S. Noda, J. D. Joannopoulos, S. P. Boyd, and S. G. Johnson, “Robust optimization of adiabatic tapers for coupling to slow-light photonic-crystal waveguides,” Opt. Express |

134. | A. Mutapcica, S. Boyd, A. Farjadpour, S. G. Johnson, and Y. Avniel, “Robust design of slow-light tapers in periodic waveguides”. Eng. Optim. |

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136. | K. W. Chun and J. Ra, “Fast block-matching algorithm by successive refinement of matching criterion,” Proc. SPIE, Vis. Commun. Image Process. , |

137. | S. G. Johnson, P. Bienstman, M. A. Skorobogatiy, M. Ibanescu, E. Lidorikis, and J. D. Joannopoulos, “Adiabatic theorem and continuous coupled-mode theory for efficient taper transitions in photonic crystals,” Phys. Rev. E |

138. | C. Manolatou, S. G. Johnson, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “High-density integrated optics,” J. Lightwave Technol. |

**OCIS Codes**

(000.3860) General : Mathematical methods in physics

(120.4570) Instrumentation, measurement, and metrology : Optical design of instruments

(130.2790) Integrated optics : Guided waves

(160.3918) Materials : Metamaterials

**ToC Category:**

Physical Optics

**History**

Original Manuscript: April 8, 2013

Revised Manuscript: May 25, 2013

Manuscript Accepted: May 28, 2013

Published: June 7, 2013

**Citation**

David Liu, Lucas H. Gabrielli, Michal Lipson, and Steven G. Johnson, "Transformation inverse design," Opt. Express **21**, 14223-14243 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-12-14223

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