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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 12 — Jun. 17, 2013
  • pp: 14223–14243
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Transformation inverse design

David Liu, Lucas H. Gabrielli, Michal Lipson, and Steven G. Johnson  »View Author Affiliations


Optics Express, Vol. 21, Issue 12, pp. 14223-14243 (2013)
http://dx.doi.org/10.1364/OE.21.014223


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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

In this work, we introduce the technique of transformation inverse design, which combines the elegance of transformation optics [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] .

8

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] .

] (TO) with the power of large-scale optimization (inverse design), enabling automatic discovery of the best possible transformation for given design criteria and material constraints. We illustrate our technique by designing multimode waveguide bends [9

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

21

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] .

] and mode squeezers [20

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

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] .

], then measuring their performance with finite element method (FEM) simulations. Most designs in transformation optics use either hand-chosen transformations [3

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

, 14

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] .

17

17. H. Xu, B. Zhang, Y. Yu, G. Barbastathis, and H. Sun, “Dielectric waveguide bending adapter with ideal transmission,” Opt. Express 29(6):1287–1290 (2012).

, 22

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

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

32

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

] (which often require nearly unattainable anisotropic materials), or quasiconformal and conformal maps [2

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

, 9

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

13

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

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

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

51

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] .

] which can automatically generate nearly-isotropic transformations (either by solving partial differential equations or by using grid generation techniques) but still require 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

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] .

]. Transformation inverse design combines elements of both transformation optics and inverse design while overcoming their limitations. First, the use of optimization allows us to incorporate arbitrary fabrication constraints while at the same time searching the correct space of transformations without unnecessarily underconstraining or overconstraining the problem. Second, instead of solving Maxwell’s equations, we require only simple derivatives to be computed at each optimization step. This is because transformation optics works by using a coordinate transformation 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] .

, 4

4. U. Leonhardt and T. Philbin, Geometry and Light: The Science of Invisibility (Dover, 2010).

, 33

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

, 40

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

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

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

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

]) and then employing transformed materials which are given in terms of the Jacobian 𝒥ij = ∂x′j/∂xi 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

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

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

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

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] .

]), and is therefore particularly attractive for designing multimode optical devices [20

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] .

, 34

34. Q. Wu, J. P. Turpin, and D. H. Werner, “Integrated photonic systems based on transformation optics enabled gradient index devices,” Light: Science and Applications 1(e38) (2012).

, 85

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

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] .

] (such as mode squeezers, expanders, splitters, couplers, and multimode bends) with no inter-modal scattering. (Similar ideas appeared even earlier in the context of electrostatic cloaking by anisotropic conductivities [74

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] .

, 75

75. A. Greenleaf, M. Lassas, and G. Uhlmann, “On nonuniqueness for Calderón’s inverse problem,” Math. Res. Letters 10(5):685–693 (2003).

].) Examples of such transformations are shown in Fig. 1.

Fig. 1 Three possible applications of transformation optics for multimode waveguides: squeezer, expander, and bend. Dark areas indicate higher refractive index.

One major difficulty with transformation optics is that most functions 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

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

90

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] .

] or naturally birefringent materials [25

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

, 30

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] .

]. However, in the infrared regime (where metals are lossy) it is far easier to instead fabricate effectively isotropic dielectric materials, provided that the refractive index falls within the given bounds nmin and nmax 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

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

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

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

, 72

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

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

, 91

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

93

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] .

] or waveguides with variable thickness [94

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

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] .

]). This requirement means that we would prefer to consider the subset of transformations that can be mapped to approximately isotropic dielectric materials.

In Sec. 2.1, we review the equations of transformation optics. In Sec. 2.2, we describe situations where the transformation-designed material can be mapped to isotropic media. In Sec. 2.3, we point out that such isotropic transformations, due to their analyticity, always have undesirable interface discontinuities when coupled into untransformed regions. In Secs. 2.4 and 2.5, we review the techniques of quasiconformal mapping (as used in both the transformation optics and mathematical analysis literature) and scalarization of nearly isotropic transformations. We show that the inherent restrictions of quasiconformal mapping can be circumvented by directly optimizing the map using transformation inverse design. In Secs. 3.1 and 3.2, we design a nearly isotropic transformation for a 90°-bend by perturbing from the highly anisotropic circular bend transformation. In Secs. 3.3 and 3.4, we set up the bend optimization problem and the spectral parameterization. In Sec. 4, we present the optimized structure, which reduces anisotropy by several orders of magnitude compared to the circular TO bend. In Sec. 4.1, we present finite element simulation results comparing our optimized design to the conventional non-TO bend and the circular TO bend. In Secs. 4.2 we show that minimizing the mean anisotropy can lead to pockets of high anisotropy (which in turn leads to greater intermodal scattering) while minimizing the peak does not. In Secs. 4.3, we discuss the tradeoff between the bend radius and the optimized anisotropy. In Sec. 5 we briefly present methods and results for applying transformation inverse design to optimize mode squeezers.

2. Mathematical preliminaries

2.1. Transformation optics

The frequency domain Maxwell equations (fields ∼ eiωt), without sources or currents, in linear isotropic dielectric media [ε = ε(x), μ = μ0] are
×H=iωε(x)E×E=iωμ0H.
(1)
Consider a coordinate transformation x′(x) with Jacobian 𝒥ij=xj'xi. We define the primed gradient vector as (x,y,z)=𝒥1 and the primed fields as E′𝒥−1E and H′𝒥−1H. 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

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

], as
×H=iωεE×E=iωμH,
(2)
where the effects of the coordinate transformation have been mapped to the equivalent tensor materials
μ=μ0𝒥T𝒥det𝒥ε=ε(x)𝒥T𝒥det𝒥.
(3)
This equivalence has become known as 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] .

]. For a further discussion of space–time transformations and connections to negative refraction, see [4

4. U. Leonhardt and T. Philbin, Geometry and Light: The Science of Invisibility (Dover, 2010).

] and [121

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

].

Most useful applications of TO require that the transformation be coupled to untransformed regions (e.g. the input and output straight waveguides in the case of a bend transformation, or the surrounding air region for the case of a ground-plane cloaking transformation). However, in order for TO to guarantee that the interface between transformed and untransformed regions be reflectionless, the transformation must be equivalent to a 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

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

, 105

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] .

], and this is in fact a necessary condition as well [103

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

]. Although a general anisotropic transformation need only have 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.

Fig. 2 The interface between the transformed and untransformed region must have x′ continuous in order for there not to be any interface reflections.

2.2. Transformations to isotropic dielectric materials

For the vast majority of transformations, the materials in Eq. (3) are anisotropic tensors. However, for certain transformations, the tensors are effectively scalar. Suppose that the transformation x′(x) is 2D (z′ = z and xz=0), making 𝒥 block-diagonal (with the zz element independent of the xy block). Then, the xy block of 𝒥T𝒥 is isotropic if and only if the diagonal elements are equal and the off-diagonal elements vanish:
|x|2|y|2=0xy=0.
(4)
In this case, the 𝒥 part of Eq. (3) becomes
𝒥T𝒥det𝒥=(111det𝒥).
(5)

This isotropy has different implications for transverse-magnetic (TM) polarized modes in 2D (which have E = Eẑ and H· ẑ = 0) versus transverse-electric (TE) polarized modes (which have E = 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
×H=iωε(x)det𝒥E
(6)
×E=iωμ0H.
(7)
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 det𝒥 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

119. J. D. Jackson, Classical Electrodynamics (Wiley, 1998).

] Ch. 8.10) and commuting 1/μ′ with one of the curls in the Maxwell equations. In particular, Eq. (2) can be written:
××E=ω2ε0μ0(det𝒥)E+𝒪(det𝒥),
(8)
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

If a transformation has an isotropic 𝒥, then the transformation preserves angles in the xy plane. Additionally, if det𝒥 > 0, then the transformation also preserves handness and orientation. The combination of these two properties is called a conformal map [100

100. W. Rudin, Real and Complex Analysis (McGraw–Hill, 1986).

], and is the only case where the situation in Sec. 2.2 can be realized. We only consider transformations with det𝒥 > 0 in order to restrict ourselves to dielectric materials. Also, a det𝒥 > 0 transformation coupled continuously to an untransformed (det𝒥 = 1) region would require singularities (det𝒥 = 0) at some points. Conformal maps are described by analytic functions, which are of the form x′ + iy′ = w′(w) (where wx + iy is the untransformed complex coordinate) and whose real and imaginary parts satisfy the Cauchy–Riemann equations of complex analysis [100

100. W. Rudin, Real and Complex Analysis (McGraw–Hill, 1986).

, 101

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

].

However, true conformal maps cannot directly be used for transformation optics in typical applications, because of the impossibility of coupling them to untransformed regions with the boundary conditions discussed in Sec. 2.1. In particular, the uniqueness theorem of analytic functions [101

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

, Thm. 10.39] tells us that if w′(w) = w in some region, then w′(w) = w everywhere (similarly for a simple rotation or translation in some regions).

2.4. Quasiconformal maps and measures of anisotropy

2.5. Scalarization errors for nearly isotropic materials

The minimum-anisotropy quasiconformal map is then scalarized (as in [33

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

]) by approximating it with an isotropic dielectric material. As shown in Sec. 2.2, a perfectly isotropic 2D transformation of a geometry with an isotropic dielectric material that guides TM modes E0, H0 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 μ′ = 𝕀 + Δμ, where the anisotropic part Δμ is proportional to 𝕂 − 1 to lowest order. While Δμ′ ≠ 0 cannot be fabricated using dielectric gradient index processes, one can neglect this small correction so that the actual fabricated material has permeability μ′approx = 𝕀. In practice, we absorb any Δμ′ into ε′ by ε′ multiplying by the average eigenvalue of μ′
μ=λ1+λ22λ1λ2=tr𝒥T𝒥2det𝒥
(12)
but this does not change the 𝒪 (𝕂 − 1) error.

A Born approximation [126

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] .

128

128. W. C. Chew, Waves and Fields in Inhomogeneous Media (IEEE, 1995).

] tells us that, given an exact transformation with no scattering, any small error of Δε and Δμ will generically lead to scattered fields with magnitudes of 𝒪(|Δε|+|Δμ|) and scattered power of 𝒪(|Δε|2 +|Δμ|2). The modes of the approximate scalarized material μ′approx, ε′approx are then the exact guided modes plus scattered power corrections of 𝒪(|Δμ|2) = 𝒪[(𝕂 − 1)2].

A similar analysis explains why we must explicitly impose continuity of 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 det𝒥 (and hence the discontinuity in the refractive index) is of order 𝒪(𝕂1), which would lead to 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 maxx 𝕂 − 1 ≈ 0.0005. Therefore, in Sec. 3.4 we impose continuity of 𝒥 explicitly.

2.6. General optimization of anisotropy

In this paper, we directly minimize 𝕂 using large-scale numerical optimization while keeping track of constraints on the transformation x′ and its Jacobian 𝒥, as well as the engineering fabrication bounds nmin and nmax. 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
minx(x)𝕂(x)subjectto{x,𝒥continuousatinput/outputinterfacesnminn(x)nmax,
(13)
where 〈𝕂(x)〉 is a functional norm taken over the domain of x′(x). We consider two possible norms: the L1 norm (the mean 〈𝕂〉x), and the L norm (maxx 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′ 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

Fig. 3 In the transformation process, the untransformed straight waveguide is bent, perturbed, and optimized. Darker regions indicate higher refractive index

3.1. Simple circular bends

First, we consider a simple circular bend transformation (which we refer to hereafter as the circular TO bend) that maps a rectangular segment of length 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 RxR+1 and L2yL2, with the untransformed segment length L=πR2 equal to the inner arclength of the bend. The transformation x′(x) can be written as
x=rcosθy=rsinθz=z,
(14)
where r = x and θ=yR. While x′ is continuous at the input/output interfaces y=±L2, one issue is that 𝒥 is not continuous there, which can be seen from det𝒥=xR1. Another issue is that μ′μ0𝕀 is highly anisotropic. The anisotropy for this transformation is 𝕂(x,y)1=x2R+R2x1, which has a peak value maxx𝕂112R2 for R ≫ 1 at the outer radius x = R+1. Note that one can instead choose r=exp(πx2L), which gives the conformal bend x+iy=exp[π2L(x+iy)]. As explained in Sec. 2.3, this map has zero anisotropy, but neither x′ nor 𝒥 are continuous at the input/output interfaces, leading to large reflections there.

3.2. Generalized bend transformations

3.3. Numerical optimization problem

Besides minimizing the objective function 𝕂, the optimization must keep track of several constraints. First, any fabrication method will bound the overall refractive index n′ to lie between some values nmin and nmax. We choose units so that the width of the transformed region is unity (RxR + 1), and consider transforming a straight waveguide of width Δw < 1. Δw 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 |xR12|<Δw2 and low in the cladding |xR12|>Δw2. For convenience, we write this refractive index as a product n(x) = n0p(x) of an overall refractive index n0 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
n(x)=εμ=n0p(x)tr𝒥T𝒥2(det𝒥)2
(19)
where the average eigenvalue μ′ of the magnetic permeability Eq. (12) has been absorbed into the dielectric index. The overall refractive-index scaling n0 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.

With these constraints, there are several ways to set up the optimization problem, depending on which norm we are minimizing. One method is to minimize the peak anisotropy maxx 𝕂 with xG 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

116. S. Boyd and L. Vandenberghe, Convex Optimization (Cambridge University, 2004).

]: we introduce a dummy variable t and indirectly minimize the peak 𝕂 using a differentiable inequality constraint between t and 𝕂(x) at all xG:
minr(x),θ(x),n0,L,ttsubjectto:{continuityconditions17,18nminnpp(x)tr𝒥T𝒥2(det𝒥)2nmaxforxGR=R0𝕂(x)tforxG.
(20)
For comparison, we explain in Sec. 4.2 why the L norm is better to minimize than the L1 norm (the mean anisotropy).

The minimization of the L1 norm, 〈𝕂〉x = ∫ 𝕂dxdy/area [which is differentiable in terms of the parameters r(x), θ(x), n0, and L] is implemented as
minr(x),θ(x),n0,L𝕂xsubjectto:{continuityconditions17,18nminn0p(x)tr𝒥T𝒥2(det𝒥)2nmaxforxGR=R0.
(21)

3.4. Spectral parameterization

To faciliate efficient computation of the objective and constraints, the functions r and θ can be written as the circular bend transformation plus perturbations parametrized in the spectral basis [115

115. J. P. Boyd, Chebyshev and Fourier Spectral Methods (Dover, 2001).

, 116

116. S. Boyd and L. Vandenberghe, Convex Optimization (Cambridge University, 2004).

]:
r(x,y)=x+,mN,NmCmrT(2x2R1)cos2mπyLθ(x,y)=πy2L+1x,mN,NmCmθT(2x2R1)sin2mπyL,
(22)
where the coordinate 2x − 2R − 1 has been centered appropriately for the domain [−1, 1] of degree- 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:
mNmCmr(1)m=0mNmCmθ(1)mm={L8πR418,=018,=10,2.
(23)
These equations are solved to simply eliminate the CNmr,θ coefficients before optimization.

This spectral parametrization has several advantages over finite-element discretizations such as the piecewise-linear parameterization of [106

106. O. Weber, A. Myles, and D. Zorin, “Computing extremal quasiconformal maps,” Symp. Geom. Process. 31(5):1679–1689 (2012).

]. First, the spectral basis converges exponentially for smooth functions [115

115. J. P. Boyd, Chebyshev and Fourier Spectral Methods (Dover, 2001).

]. We found that only a small number (N × Nm < 100) of spectral coefficients Cr,θ are needed to achieve very low-anisotropy (𝕂−1 ≈ 10−4) 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 Nm.

With this spectral parameterization, the formulation of the optimization problem Eq. (20) becomes
min{Cmr,θ},n0,L,ttsubjectto:{constraint23nminn0p(x)tr𝒥T𝒥2(det𝒥)2nmaxforxGR=R0𝕂(x)tforxG.
(24)

The local optimization was performed using the derivative-free COBYLA non-linear optimization algorithm [129

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

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

] in the NLopt package [131

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

]. In principle, we can make the optimization faster by analytically computing the derivatives of the objective and constraints with respect to the design parameters and using a gradient-based optimization algorithm, but that is not necessary because both tr𝒥T𝒥 and det𝒥, 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

A min||𝕂|| 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, Nm = 8. The objective and constraints were evaluated on a 100 × 140 grid G in x (Chebyshev points in the x direction and a uniform grid in the y direction). This design had maxx 𝕂 − 1 ≈ 5 × 10−4 and mean 〈𝕂〉 − 1 ≈ 10−4. In comparison, the circular TO bend of the same radius has maxx 𝕂 − 1 ≈ 0.1 and 〈𝕂〉 − 1 ≈ 10−2.

Fig. 4 Optimization decreases anisotropy by a factor of 10−4, while dramatically improving the scattered-power matrix.

The 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] .

]), with the conventional non-TO bend [simply bending the waveguide profile around a circular arc with 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 y=L2, and the scattered-power matrix T was computed using the measured fields at the output interface y=L2. The scattered-power matrix is defined as
Tij=|RR+1dxθ^(Ej0×Hi)|y=L22,
(25)
where −θ̂ is the propagation direction of the guided modes, Ej0 is the normalized electric field of the jth exactly guided mode of the non-scalarized material (μ′, ε′), and Hi is the actual magnetic field of the approximate scalarized material at the interface after injecting a normalized mode Ei0 at the input interface. This makes Tij equal to the power scattered into the jth output mode from the ith 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.]

The electric-field profiles for the fundamental mode, displayed in Fig. 5, show a dramatic difference in the performance of the optimized structure versus the other structures. Both the conventional and circular TO bend show heavy intermodal scattering in the bend region, while the optimized transformation displays very little scattering.

Fig. 5 FEM field profiles show heavy scattering in the conventional non-TO and scalarized circular bends, but very little scattering in the optimized bend.

4.2. Minimizing max versus minimizing mean

We found a clear difference between minimizing the peak anisotropy versus minimizing the mean. The results of an optimization run with R = 2.5, N = 3, and Nm = 6 are shown in Fig. 6. Both structures had very low mean anisotropy 〈𝕂〉x − 1. The mean-minimized structure, at 〈𝕂〉 − 1 ≈ 10−5, 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 maxx 𝕂 − 1 ≈ 2 × 10−4 as opposed to maxx 𝕂 − 1 ≈ 5 × 10−4 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 maxx 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 T44 = 0.89 for the fourth mode).

Fig. 6 Anisotropy profile and scattered-power matrices for optimized designs that minimize the mean and the peak, with R = 2.5, N = 3, and Nm = 6.

4.3. Tradeoff between anisotropy and radius

Fig. 7 Successive optimization with N = 5, Nm = 8 results in a power law decaying trade-off maxx 𝕂 − 1 ∼ R−4 at low R and an exponentially decaying tradeoff at higher R. For comparison, the unoptimized anisotropy for the circular TO bend is shown above.

For 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] .

136

136. K. W. Chun and J. Ra, “Fast block-matching algorithm by successive refinement of matching criterion,” Proc. SPIE, Vis. Commun. Image Process. , 1818:552–560 (1992).

], in which optima for smaller Nℓ,m are used as starting points for optimizing using larger Nℓ,m)

5. Mode squeezer

We also applied transformation inverse design to another interesting geometry: a mode squeezer that concentrates modes and their power in a small region in space, again with minimal intermodal scattering (quite unlike a conventional lens, which is intrinsically angle/mode-dependent), similar to the problem considered in [22

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] .

] (which did not construct isotropic designs). We choose the untransformed region to be −1 ≤ x ≤ 1 and 0 ≤ yL. The goal of this transformation x′(x) is to focus the beam by minimizing the mid-beam width
W=11dx(xx)2+(yx)2|y=L2.
(26)
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
x(x,y)=x+,mN,NmCmxT(x)sin(2m+1)πyL
(27)
y(x,y)=y+,mN,NmCmyT(x)sin(2m+1)πyL,
(28)
The sine series automatically satisfies mirror symmetry about y=L2 and continuity of x′ at the input/output interfaces y = 0, L. However, we found that constraining the coefficients Cx,y to enforce continuity of 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.

Finite-element Maxwell simulations, shown in Fig. 8, demonstrate that the optimized design is greatly superior to a simple Gaussian taper transformation designed by hand. The Gaussian transformation was given by x(x)=xxαexp[β(yL2)2], where 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] .

]. However, the optimized TO design is much too short to be in this adiabatic regime. If it were in the adiabatic regime, then taking the same design and simply stretching the index profile to be more gradual (a taper twice as long) would reduce the scattering, but in Fig. 8 we perform precisely this experiment and find that the stretched design increases the scattering.

Fig. 8 Optimized squeezer outperforms gaussian taper and stretched optimized squeezers in finite element simulations.

6. Concluding remarks

Appendix

In this Appendix, we briefly derive the fact, mentioned in Sec. 2.5, that the endfacet discontinuity scales much worse with anisotropy than the scalarization errors in the transformation interior, which leads us to impose an explicit continuity constraint on the Jacobian 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
𝒥=(10δ1+Δ),
(29)
where δxy and Δyy1 are small quantities (≪ 1) if 𝒥T𝒥 is nearly isotropic. The anisotropy Eq. (11) is then:
𝕂1=1+δ2+(1+Δ)22(1+Δ)1=12(δ2+Δ2)+𝒪(δ2Δ+Δ3).
(30)
The determinant then satisfies
det𝒥1=Δ
(31)
=2(𝕂1)δ2+𝒪(δ2Δ+Δ3)
(32)
=𝒪(𝕂1).
(33)
This square-root dependence is also reflected in the refractive index n=εμ and leads to 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

This work was supported in part by the AFOSR MURI for Complex and Robust On-chip Nanophotonics (Dr. Gernot Pomrenke), grant number FA9550-09-1-0704.

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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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