OSA's Digital Library

Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 18 — Aug. 29, 2011
  • pp: 16821–16829
« Show journal navigation

Scaling two-dimensional photonic crystals for transformation optics

Zixian Liang and Jensen Li  »View Author Affiliations


Optics Express, Vol. 19, Issue 18, pp. 16821-16829 (2011)
http://dx.doi.org/10.1364/OE.19.016821


View Full Text Article

Acrobat PDF (1387 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

Abstract

We propose a method to manipulate Bloch waves in curved photonic crystals for achieving transformation optical devices in two dimensions. Instead of starting from an effectively homogeneous medium, we transform a regular photonic crystal into a curved one in the physical space. A scaling law is established to construct the curved photonic crystal with similar unit cells and different scales, which is made of dielectrics only. A wave compressor and a bending waveguide are designed using dielectrics with indices only from 1 to 4. The approach will be useful in constructing low-loss transformation media requiring small indices, or large anisotropy which is particularly difficult for E-polarization using the conventional effective medium approach.

© 2011 OSA

1. Introduction

Transformation optics (TO) has been recently established as a new method in optical design. In particular, it makes it possible to demonstrate some very novel applications, such as an invisibility cloak [1

1. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006). [CrossRef] [PubMed]

,2

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

]. Many different optical devices have been theoretically designed as well [1

1. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006). [CrossRef] [PubMed]

20

20. Y. Liu, T. Zentgraf, G. Bartal, and X. Zhang, “Transformational plasmon optics,” Nano Lett. 10(6), 1991–1997 (2010). [CrossRef] [PubMed]

]. Most of the TO devices can be achieved with anisotropic transformation media, which are in turn realized through metamaterials. Because of the ease of fabrication, most of the experimental studies were performed in the microwave regime using metamaterials with artificial metallic resonating atoms [21

21. R. Liu, 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]

25

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

]. On the other hand, demonstrations at much smaller wavelengths are actually quite challenging because of the stronger absorption of metallic elements and the larger difficulty in fabricating macroscopic samples with subwavelength building blocks. To tackle these problems, there are approaches involving simplification of the transformation mappings (with reduced or limited functionalities). This leads to transformation media with smaller index variations and constant anisotropy. Cloaking at infrared and at visible frequencies has been successfully demonstrated with this approach [26

26. J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8(7), 568–571 (2009). [CrossRef] [PubMed]

32

32. J. Fischer, T. Ergin, and M. Wegener, “Three-dimensional polarization-independent visible-frequency carpet invisibility cloak,” Opt. Lett. 36(11), 2059–2061 (2011). [CrossRef] [PubMed]

]. More recently, photonic crystals (PhCs) have been proposed as an alternative means to construct transformation optical devices within the eikonal limit. A one-dimensional photonic crystal from different dielectrics is proposed to approximate the effective medium of a TO invisibility cloak [33

33. Y. A. Urzhumov and D. R. Smith, “Transformation optics with photonic band gap media,” Phys. Rev. Lett. 105(16), 163901 (2010). [CrossRef] [PubMed]

]. Although PhCs technically work in the frequency regime with wavelengths comparable to the lattice constant instead of the effective medium regime, they are found reliable in many different situations where the refractive index plays a more dominant role than the impedance. For example, we can also use PhC to mimic metamaterials for negative refraction, to construct zero-averaged-index photonic band gaps, and very recently a homogenized zero refractive index material [34

34. C. Luo, S. Johnson, J. Joannopoulos, and J. Pendry, “All-angle negative refraction without negative effective index,” Phys. Rev. B 65(20), 201104 (2002). [CrossRef]

37

37. X. Huang, Y. Lai, Z. H. Hang, H. Zheng, and C. T. Chan, “Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials,” Nat. Mater. 10(8), 582–586 (2011). [CrossRef] [PubMed]

]. Because the unit cell is large with respect to the wavelength, the sample can be potentially made larger, making fabrication easier. Moreover, wave propagation within the media has a very low loss without employing resonating metallic artificial atoms. The PhC approach, together with its associated wave phenomena, in approximating a low-loss effective medium, is thus worthwhile for further exploration. It also has the potential to enrich the design paradigm of integrated optical circuits with PhCs [38

38. J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton University Press, 2008).

]. In this paper, we propose a transformation approach based on a PhC to achieve TO devices. Instead of searching for different PhCs to give a range of different equifrequency contours (EFC) in approximating a specific effective medium profile of a TO device [33

33. Y. A. Urzhumov and D. R. Smith, “Transformation optics with photonic band gap media,” Phys. Rev. Lett. 105(16), 163901 (2010). [CrossRef] [PubMed]

], our approach directly generates the TO device by transforming a regular PhC to a curved and gradient PhC. Once the function of the regular PhC is prescribed, e.g., a specific ray trajectory, a wavefront modification, etc., the curved PhC allows wave propagation in a geometrical transformed fashion. Thus, it acts as a TO device to control the propagation of Bloch waves inside. An additional advantage is that we are no longer restricted by the conventional elliptical or hyperbolic-type EFCs given by an effective medium. In the following sections, we will demonstrate our approach on a single unit cell, then on the whole PhC, and finally demonstrate two TO devices, a wave compressor and a bending waveguide.

2. Transforming photonic crystals

First, we use a schematic diagram to reveal our approach in obtaining a transformation optical device with a curved PhC. Figure 1(a)
Fig. 1 Transforming (a) a regular PhC in virtual space to (b) a curved PhC in physical space. The black dashed lines outline the boundary of the unit cells of the PhC; the black solid lines outline specific structures within each unit cell. The blue arrows indicate the wave propagation paths.
shows the regular PhC in the virtual space before transformation. The dashed lines denote the unit cell boundaries, while the solid lines represent the specific structure within each unit cell. A blue arrow is also drawn to represent a Bloch wave traveling with a fixed wave vector within the crystal. After the regular PhC is transformed to a curved one in Fig. 1(b), the Bloch wave is traveling along a bent trajectory and is actually governed by the curvature of the transformed PhC. In other words, we are controlling the Bloch waves instead of the plane waves in the effective medium regime.

The regular PhC is transformed to the curved PhC through a global mapping in one step. However, it is easier to start our discussion at the level of a single unit cell. Figure 2(a)
Fig. 2 Transforming (a) a regular unit cell to (b) the curved unit cell. The lattice constant of the regular PhC unit cell is a. The curved PhC unit cell (composed of isotropic dielectric materials) spans an angle of Δθ=π/3 with an inner radius r1=0.455a and an outer radius r2=1.455a. The curved unit cell has a circular cylinder with radius rd=0.2a and permittivity 12.5εb in background permittivityεb=1. The color map shows the E-field pattern at a normalized frequency ωa/(2πc)=0.59.
shows a single unit cell of lattice constant a in the regular PhC in Cartesian coordinate (x',y',z'). It is transformed to a curved unit cell in cylindrical coordinate (r,θ,z), shown in Fig. 2(b). In the current example, the coordinate mapping (concerning a particular unit cell) can be written as

x'=g(r),y'=Rθ,z'=z,
(1)

which is general enough for our consideration. Suppose we concentrate on the E-polarization (E-field pointing out of plane) and restrict ourselves to a dielectrics-only PhC ε(r,θ) in the physical space for ease of fabrication. According to the TO theory, the coordinate mapping induces a relationship between the regular PhC (μx'(x',y'),μy'(x',y'),εz'(x',y')) and the curved PhC ε(r,θ) through

μx'(x',y')=1μy'(x',y')=g(r)rR,εz'(x',y')=r/Rg(r)ε(r,θ).
(2)

As an example, we choose the curved unit cell as a cylindrical sector spanning an angle of Δθ=π/3 with inner radius r1=0.455a and outer radius r2=1.455a. It has a background permittivity εb=1 and consists of a circular cylinder of radius rd=0.2a and permittivity 12.5εb in the middle of the unit cell. The material parameters of the regular (square) unit cell can be obtained through Eq. (2) with R=0.955a and g(r)=r (this choice of mapping is just for ease of demonstration). The geometric parameters are chosen to satisfy RΔθ=a so that it is a square unit cell in the virtual space. The curved unit cell holds the simpler materials, i.e., dielectrics, while the regular unit cell is square in shape (easier for analysis) but with anisotropic materials in general. We also note that the shape of the cylinder inside the regular unit cell is not circular anymore. Under such a coordinate transformation, the regular and the curved unit cells should be equivalent to each other under TO. In particular, for a same set of Floquet boundary conditions applying on the two pairs of edges, the two unit cells give the same set of eigenmode frequencies (band structure). The color map in Fig. 2 shows the E-field pattern of the fourth mode for zero phase change in the radial direction (or x'-direction in virtual space) and a π/3 phase change in the angular direction (or y'-direction in the virtual space). The two eigenmodes are calculated by using the finite-element method with the commercial package COMSOL Multiphysics. Both of them appear at the same normalized frequency ωa/(2πc)=0.59, while the field patterns are just equivalent to each other according to the coordinate transformation in Eq. (1). This equivalence allows us to define the EFCs (i.e., the refractive indices, which control how light propagates in the virtual space) in the virtual space and also the effective indices in the physical space after transformation.

Next, we describe the transformation of the whole PhC starting from the established transformation of a single unit cell. There are many different ways to transform the periodic PhC into the curved PhC in physical space. We therefore need to generate a family of different unit cells in the physical space that are equivalent to the same unit cell in the virtual space. Here, our approach is to simultaneously apply geometric scaling and permittivity scaling to a single unit cell in the physical space by

ωΔrεb2πc=constant,
(3)

which can be regarded as the scaling law of the curved photonic crystal and can be proved from TO theory with geometrical scaling. Therefore, the resultant curved PhC is periodic in the angular direction, while in the radial direction each ring of thickness Δr is geometrically scaled down by a constant factor s=r2/r1, and εb is scaled up by a constant factor s2 from its neighboring outer ring. Based on this principle, we can design the TO devices through the scaling of the curved unit cells. The global mapping (written as g(r) again) that corresponds to this scaling process can be written as g(r)=R/r, where RΔθ=a to ensure our square regular unit cell.

Furthermore, the above analysis can also be easily adapted to the H-polarization for dielectric-only PhCs by

εx'(x',y')ε(r,θ)=ε(r,θ)εy'(x',y')=g(r)rR,μz'(x',y')=r/Rg(r)
(4)

For the scaling law, technically speaking, we should scale the magnetic permeability from TO theory. However, we scale the permittivity, instead of the magnetic permeability, by the same amount. This has the advantage that the EFC still remains unchanged, which is a requirement for how the wave propagates, but the whole curved PhC still consists of only dielectrics for ease of fabrication. Therefore, we use the scaling law specified by Eq. (3) for both polarizations, but we have to keep in mind that there is such a reduced-parameter approximation in the H-polarization. Although we have discussed the scaling law according to the mapping in Eq. (1), the scaling law can also be used in other kinds of mappings. For example, in conformal mappings of moderate curvatures, different unit cells in the physical space can be regarded as similar in shape, and the materials inside can be obtained through the scaling law, except that Δr now generally means the size of each cell.

3. Transformation optical devices with curved PhC

Now, we can use the scaling approach to design transformation optical devices. We begin with the bending waveguide, which has been investigated by other transformation approaches with effective media [10

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

12

12. W. X. Jiang, T. J. Cui, X. Y. Zhou, X. M. Yang, and Q. Cheng, “Arbitrary bending of electromagnetic waves using realizable inhomogeneous and anisotropic materials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 78(6), 066607 (2008). [CrossRef] [PubMed]

]. It serves as a simple example (with the schematic also shown in Fig. 1) to demonstrate how a curved PhC with scaling law can be used for wave manipulation. Figure 3 (a)
Fig. 3 (a) A curved PhC bending waveguide with inner radius 3.66a and outer radius 6.87a. Each unit cell of the outermost layer spans an angle of Δθ=9o with thickness a in the r-direction. It consists of a circular cylinder with radius rd=0.275a and permittivity 4εb in the background permittivity εb=1. The scale factor is s=1.17. (b) EFC (black line) of the regular PhC at normalized frequency ωa/(2πc)=0.686. The red dashed line represents the EFC of the corresponding isotropic effective medium. (c) H-field pattern at the same normalized frequency for a plane wave incident on the upper port. (d) H-field pattern for the corresponding effective medium εeff=nrega2/(rΔθ)2and μeff=nreg, where nreg=0.146. All length scales are normalized to a.
shows a bending waveguide (for H-polarization) constructed by a curved PhC in which a Bloch wave can make a turn of 180 degrees. The whole waveguide consists of three sections enclosed by perfect electric conductors (PEC). The initial and final sections are two rectangular air-filled waveguides with height 3.21a. The waveguide in between is bounded by two half circles of radii 3.66a and 6.87a and is filled with the curved PhC of four layers of unit cells. Each unit cell of the outermost layer has a thickness of a in the radial direction and spans an angle of Δθ=9o. It consists of a circular cylinder of radius 0.275a and permittivity 4εb in a background permittivity εb=1. The unit cells in the inner three layers are consecutively geometrically scaled down by a constant factor s=1.17 with permittivities scaled up by s2 according to our scheme. The index distribution of the whole bending waveguide is shown in Fig. 3(a), ranging from 1 to 3.2. Based on Eqs. (1) and (4), the geometric and constitutive parameters of the virtual regular unit cell can be derived. The working frequency is chosen at ωa/(2πc)=0.686. The EFC at this frequency is shown as the black solid line in Fig. 3 (b), which can approximate a circle (red dashed lines), so that the virtual regular PhC can approximate an effectively isotropic medium. We note that our material (PhC) design comes from transforming physical to virtual space, while the device design comes from transforming virtual to physical space. An input plane wave is then fed into the upper port of the waveguide with the curved PhC, and the simulated H-field pattern is shown in Fig. 3 (c); the plane wave couples to the Bloch wave inside the curved PhC, propagates along a bent path to turn 180 degrees, and exits at the lower port with high transmission and an elapsed phase about 4π.

To gain further understanding of the curved PhC, we investigate its effective medium representation. The size of the dispersion surface can be fitted by ka=nregωa/c, giving rise to nreg=0.146. The minus sign indicates a negative group velocity (the size of the EFC decreases with an increase of frequency). However, only the magnitude but not the minus sign is relevant to our investigation here, since we are exploring only the normal incidence but not the effect of negative refraction (which can be potentially useful in future investigations). We have also chosen a case in which there is nearly no reflection from the regular PhC so that we can simply set the effective impedance to Zreg=1 as an approximation. It corresponds to an effective medium μz'=εx'=εy'=nreg, which the regular PhC is mimicking. It induces, through TO with Eq. (1) and g(r)=R/r, the effective medium in the physical space to be governed by μz(eff)=nrega2/(rΔθ)2 and εr(eff)=εθ(eff)=nreg. However, since only the permittivity of the curved PhC is scaled as mentioned, the effective medium is therefore governed by

εeff=nrega2/(rΔθ)2,μeff=nreg.
(5)

The same waveguide is then simulated again but with the effective medium instead of the curved PhC. The result shown in Fig. 3 (d) with the effective medium agrees very well with the simulation with the curved PhC with a total phase elapse around 4π at the U-turn and with high transmission. For both simulations, the wavefronts at the entrance and exit of the waveguides are not completely flat owing to the approximation we introduced for H-polarization, that is, that we scale the permittivity instead of the magnetic permeability. As a result, we can conclude that the curved PhC in Fig. 3 (a) can mimic the gradient index profile of a bending waveguide.

An advantage of transforming PhCs, instead of effective media, is that more general shapes for the EFCs are available apart from ellipses and hyperbolas. Here, we would like to use a curved PhC to design a wave compressor [13

13. 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]

] by exploring the use of a PhC to mimic a very anisotropic effective medium, for E-polarization in our case. This is particular difficult in the effective medium approach without using artificial magnetic resonances, which are usually associated with high absorption loss. Figure 4 (a)
Fig. 4 (a) Transforming a regular PhC with flat EFC to achieve a wave compressor. (b) Index profile of the wave compressor. The compressor has inner radius R1=10.5a and outer radius R2=19.6a; see text for details of the configuration. (c) EFC of the regular PhC at normalized frequency ωa/(2πc)=0.58. (d) E-field pattern when a Gaussian beam of width 10.3a is incident from the left. The beam is compressed to a width of 5.5a at the exit. (e) E-filed pattern of the same compressor with the corresponding effective medium, εeff=nrega2/(rΔθ)2, μθ(eff)=nreg, and μr(eff)=, where nreg=0.261. (f) The normalized E-field amplitude distribution of the beam along the y-direction. The blue solid line represents the distribution of the incident beam. The black dashed line represents the ideal exit beam profile with compressed ratio R2/R1, and the red solid line represents the actual distribution of the exit beam from the simulation. All length scales are normalized to a.
shows the working principle of the wave compressor. Here, we transform a regular PhC with very flat EFC, or equivalently very large anisotropy, although the EFC is not an ellipse anymore. The Bloch waves inside are propagating only along the horizontal direction for a range of different incidence angles θ due to the flat EFC. The curved PhC (circular in shape) with inner radius R1 and outer radius R2 is also shown in the same figure. The Bloch waves inside can only travel along the radial direction. By choosing a permittivity (R2/R1)2 for the output medium, every incident horizontal ray to the curved PhC also exits in the horizontal direction (blue arrows). This is due to the conservation of angular momentum for the traveling wave, similar to an electromagnetic hyperlens. Because this picture holds for every horizontal ray, the device thus compresses a wide beam to a narrow beam, acting like a wave compressor.

Figure 4(b) shows a closer look at the wave compressor. The curved PhC is bounded by two circles of radii R1=10.5a and R2=19.6a and is filled with 12 layers of unit cells. Each unit cell in the outermost layer has a thickness a in the radial direction and spans an angle of Δθ=3o. It consists of a cylinder in a shape of a small sector (looks like a bar in Fig. 4 (b), has thickness 0.6a in radial direction and spans an angle of 0.9°) of permittivity 5εb in a background permittivity εb=1. The inner 11 layers are consecutively geometrically scaled down by s=1.05 with permittivities scaled up by s2 according to our scheme. The color map shows the refractive index profile ranging from 1 to 4. Again, we transform the unit cell at the outermost layer to the regular unit cell and solve for the EFC, which is shown in Fig. 4(c) at a normalized frequency ωa/(2πc)=0.58 for the E-polarization. We have chosen such a configuration because its flatness of EFC and also because of the very small reflection for the regular PhC; i.e., it is impedance matched to vacuum. We can therefore assign an effective medium εz'=μy'=nreg and μx'= where nreg=0.261, which the regular PhC is mimicking. Now, a Gaussian beam of width 10.3a impinges (from the left hand side) on the wave compressor horizontally. Figure 4 (d) shows the simulated E-field pattern. The red-dashed lines indicate the edge of the beam profile, as predicted by the ray picture in Fig. 4(a). The agreement is very good, showing that the beam is now compressed in the vertical direction. We emphasize that the regular PhC is transformed to the curved PhC directly without the need of an effective medium representation in the physical space. However, we can still work out the effective medium and investigate how faithfully the curved PhC is mimicking it. From TO theory with Eq. (1) and g(r)=R/r, the effective medium in the physical space is induced form the one in the virtual space and is governed by

εeff=nrega2/(rΔθ)2,μr(eff)=,μθ(eff)=nreg.
(6)

The same compressor is then simulated again with the effective medium instead of the curved PhC. The E-field profile is shown in Fig. 4(e), which agrees very well with the simulation with the curved PhC; even the phase information in the compressor looks almost the same as the one in Fig. 4(d). According to the discussion of Fig. 4(a), the wave compressor has a compression ratio R2/R1 for the size of the beam. Figure 4(f) compares the beam width of the exit beam along the vertical direction. The blue solid line shows the normalized E-field amplitude distribution of the incident beam for comparison. The black dashed line shows the ideal exit beam if it is compressed by a ratio of R2/R1=1.87. The red solid line is the normalized E-filed amplitude distribution of the exit beam from the simulation with the curved PhC, which is nearly the same as the ideal case. Since the beam width is compressed, the power of the beam will be enhanced with the same ratio. We have calculated the power at the center of the exit beam, it is enhanced by a ratio around 1.87 also. Therefore the very anisotropic curved PhC in this example works as a wave compressor.

4 Conclusion

In this work, transformation optical devices can be designed by transforming a regular photonic crystal to a curved and gradient one. We established a scaling law for curved photonic crystals, which can be used to construct the whole transformation medium. The approach will be useful in constructing transformation media requiring refractive indices smaller than 1, being negative or highly anisotropic, which can be made from low-loss dielectric-only photonic crystals. In particular, we can construct a low-loss very anisotropic medium for E-polarization, which is difficult using conventional effective medium. A bending waveguide and a wave compressor are thus demonstrated in order to control the propagation of Bloch waves.

Acknowledgment

We acknowledge financial support of this work from City University of Hong Kong (SRG Project Nos. 7200168 and 7008079).

References and Links

1.

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006). [CrossRef] [PubMed]

2.

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

3.

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]

4.

A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, “Electromagnetic wormholes and virtual magnetic monopoles from metamaterials,” Phys. Rev. Lett. 99(18), 183901 (2007). [CrossRef] [PubMed]

5.

W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photonics 1(4), 224–227 (2007). [CrossRef]

6.

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(6), 063903 (2008). [CrossRef] [PubMed]

7.

M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell’s equations,” Photonics Nanostruct. Fundam. Appl. 6(1), 87–95 (2008). [CrossRef]

8.

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

9.

V. M. Shalaev, “Physics. Transforming light,” Science 322(5900), 384–386 (2008). [CrossRef] [PubMed]

10.

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

11.

J. Huangfu, S. Xi, F. Kong, J. Zhang, H. Chen, D. Wang, B.-I. Wu, L. Ran, and J. A. Kong, “Application of coordinate transformation in bent waveguides,” J. Appl. Phys. 104(1), 014502 (2008). [CrossRef]

12.

W. X. Jiang, T. J. Cui, X. Y. Zhou, X. M. Yang, and Q. Cheng, “Arbitrary bending of electromagnetic waves using realizable inhomogeneous and anisotropic materials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 78(6), 066607 (2008). [CrossRef] [PubMed]

13.

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]

14.

Y. Lai, H. Chen, Z. Q. Zhang, and C. T. Chan, “Complementary media invisibility cloak that cloaks objects at a distance outside the cloaking shell,” Phys. Rev. Lett. 102(9), 093901 (2009). [CrossRef] [PubMed]

15.

D. A. Genov, S. Zhang, and X. Zhang, “Mimicking celestial mechanics in metamaterials,” Nat. Phys. 5(9), 687–692 (2009). [CrossRef]

16.

E. E. Narimanov and A. V. Kildishev, “Optical black hole: broadband omnidirectional light absorber,” Appl. Phys. Lett. 95(4), 041106 (2009). [CrossRef]

17.

S. Liu, L. Li, Z. Lin, H. Y. Chen, J. Zi, and C. T. Chan, “Graded index photonic hole: analytical and rigorous full wave solution,” Phys. Rev. B 82(5), 054204 (2010). [CrossRef]

18.

H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Mater. 9(5), 387–396 (2010). [CrossRef] [PubMed]

19.

P. A. Huidobro, M. L. Nesterov, L. Martín-Moreno, and F. J. García-Vidal, “Transformation optics for plasmonics,” Nano Lett. 10(6), 1985–1990 (2010). [CrossRef] [PubMed]

20.

Y. Liu, T. Zentgraf, G. Bartal, and X. Zhang, “Transformational plasmon optics,” Nano Lett. 10(6), 1991–1997 (2010). [CrossRef] [PubMed]

21.

R. Liu, 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]

22.

Y. G. Ma, C. K. Ong, T. Tyc, and U. Leonhardt, “An omnidirectional retroreflector based on the transmutation of dielectric singularities,” Nat. Mater. 8(8), 639–642 (2009). [CrossRef] [PubMed]

23.

H. F. Ma and T. J. Cui, “Three-dimensional broadband ground-plane cloak made of metamaterials,” Nat Commun 1(3), 21 (2010). [CrossRef] [PubMed]

24.

Q. Cheng, T. J. Cui, W. X. Jiang, and B. G. Cai, “An omnidirectional electromagnetic absorber made of metamaterials,” N. J. Phys. 12(6), 063006 (2010). [CrossRef]

25.

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

26.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8(7), 568–571 (2009). [CrossRef] [PubMed]

27.

L. H. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nat. Photonics 3(8), 461–463 (2009). [CrossRef]

28.

J. C. Halimeh, T. Ergin, J. Mueller, N. Stenger, and M. Wegener, “Photorealistic images of carpet cloaks,” Opt. Express 17(22), 19328–19336 (2009). [CrossRef] [PubMed]

29.

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]

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] [PubMed]

31.

B. Zhang, Y. Luo, X. Liu, and G. Barbastathis, “Macroscopic invisibility cloak for visible light,” Phys. Rev. Lett. 106(3), 033901 (2011). [CrossRef] [PubMed]

32.

J. Fischer, T. Ergin, and M. Wegener, “Three-dimensional polarization-independent visible-frequency carpet invisibility cloak,” Opt. Lett. 36(11), 2059–2061 (2011). [CrossRef] [PubMed]

33.

Y. A. Urzhumov and D. R. Smith, “Transformation optics with photonic band gap media,” Phys. Rev. Lett. 105(16), 163901 (2010). [CrossRef] [PubMed]

34.

C. Luo, S. Johnson, J. Joannopoulos, and J. Pendry, “All-angle negative refraction without negative effective index,” Phys. Rev. B 65(20), 201104 (2002). [CrossRef]

35.

V. Mocella, S. Cabrini, A. S. P. Chang, P. Dardano, L. Moretti, I. Rendina, D. Olynick, B. Harteneck, and S. Dhuey, “Self-collimation of light over millimeter-scale distance in a quasi-zero-average-index metamaterial,” Phys. Rev. Lett. 102(13), 133902 (2009). [CrossRef] [PubMed]

36.

S. Kocaman, R. Chatterjee, N. C. Panoiu, J. F. McMillan, M. B. Yu, R. M. Osgood, D. L. Kwong, and C. W. Wong, “Observation of zeroth-order band gaps in negative-refraction photonic crystal superlattices at near-infrared frequencies,” Phys. Rev. Lett. 102(20), 203905 (2009). [CrossRef] [PubMed]

37.

X. Huang, Y. Lai, Z. H. Hang, H. Zheng, and C. T. Chan, “Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials,” Nat. Mater. 10(8), 582–586 (2011). [CrossRef] [PubMed]

38.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton University Press, 2008).

OCIS Codes
(160.5298) Materials : Photonic crystals
(260.2710) Physical optics : Inhomogeneous optical media
(230.3205) Optical devices : Invisibility cloaks

ToC Category:
Photonic Crystals

History
Original Manuscript: May 24, 2011
Revised Manuscript: June 27, 2011
Manuscript Accepted: July 14, 2011
Published: August 15, 2011

Citation
Zixian Liang and Jensen Li, "Scaling two-dimensional photonic crystals for transformation optics," Opt. Express 19, 16821-16829 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-18-16821


Sort:  Author  |  Year  |  Journal  |  Reset  

References

  1. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006). [CrossRef] [PubMed]
  2. U. Leonhardt, “Optical conformal mapping,” Science 312(5781), 1777–1780 (2006). [CrossRef] [PubMed]
  3. 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]
  4. A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, “Electromagnetic wormholes and virtual magnetic monopoles from metamaterials,” Phys. Rev. Lett. 99(18), 183901 (2007). [CrossRef] [PubMed]
  5. W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photonics 1(4), 224–227 (2007). [CrossRef]
  6. 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(6), 063903 (2008). [CrossRef] [PubMed]
  7. M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell’s equations,” Photonics Nanostruct. Fundam. Appl. 6(1), 87–95 (2008). [CrossRef]
  8. J. Li and J. B. Pendry, “Hiding under the carpet: a new strategy for cloaking,” Phys. Rev. Lett. 101(20), 203901 (2008). [CrossRef] [PubMed]
  9. V. M. Shalaev, “Physics. Transforming light,” Science 322(5900), 384–386 (2008). [CrossRef] [PubMed]
  10. N. I. Landy and W. J. Padilla, “Guiding light with conformal transformations,” Opt. Express 17(17), 14872–14879 (2009). [CrossRef] [PubMed]
  11. J. Huangfu, S. Xi, F. Kong, J. Zhang, H. Chen, D. Wang, B.-I. Wu, L. Ran, and J. A. Kong, “Application of coordinate transformation in bent waveguides,” J. Appl. Phys. 104(1), 014502 (2008). [CrossRef]
  12. W. X. Jiang, T. J. Cui, X. Y. Zhou, X. M. Yang, and Q. Cheng, “Arbitrary bending of electromagnetic waves using realizable inhomogeneous and anisotropic materials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 78(6), 066607 (2008). [CrossRef] [PubMed]
  13. 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]
  14. Y. Lai, H. Chen, Z. Q. Zhang, and C. T. Chan, “Complementary media invisibility cloak that cloaks objects at a distance outside the cloaking shell,” Phys. Rev. Lett. 102(9), 093901 (2009). [CrossRef] [PubMed]
  15. D. A. Genov, S. Zhang, and X. Zhang, “Mimicking celestial mechanics in metamaterials,” Nat. Phys. 5(9), 687–692 (2009). [CrossRef]
  16. E. E. Narimanov and A. V. Kildishev, “Optical black hole: broadband omnidirectional light absorber,” Appl. Phys. Lett. 95(4), 041106 (2009). [CrossRef]
  17. S. Liu, L. Li, Z. Lin, H. Y. Chen, J. Zi, and C. T. Chan, “Graded index photonic hole: analytical and rigorous full wave solution,” Phys. Rev. B 82(5), 054204 (2010). [CrossRef]
  18. H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Mater. 9(5), 387–396 (2010). [CrossRef] [PubMed]
  19. P. A. Huidobro, M. L. Nesterov, L. Martín-Moreno, and F. J. García-Vidal, “Transformation optics for plasmonics,” Nano Lett. 10(6), 1985–1990 (2010). [CrossRef] [PubMed]
  20. Y. Liu, T. Zentgraf, G. Bartal, and X. Zhang, “Transformational plasmon optics,” Nano Lett. 10(6), 1991–1997 (2010). [CrossRef] [PubMed]
  21. R. Liu, 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]
  22. Y. G. Ma, C. K. Ong, T. Tyc, and U. Leonhardt, “An omnidirectional retroreflector based on the transmutation of dielectric singularities,” Nat. Mater. 8(8), 639–642 (2009). [CrossRef] [PubMed]
  23. H. F. Ma and T. J. Cui, “Three-dimensional broadband ground-plane cloak made of metamaterials,” Nat Commun 1(3), 21 (2010). [CrossRef] [PubMed]
  24. Q. Cheng, T. J. Cui, W. X. Jiang, and B. G. Cai, “An omnidirectional electromagnetic absorber made of metamaterials,” N. J. Phys. 12(6), 063006 (2010). [CrossRef]
  25. N. Kundtz and D. R. Smith, “Extreme-angle broadband metamaterial lens,” Nat. Mater. 9(2), 129–132 (2010). [CrossRef] [PubMed]
  26. J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8(7), 568–571 (2009). [CrossRef] [PubMed]
  27. L. H. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nat. Photonics 3(8), 461–463 (2009). [CrossRef]
  28. J. C. Halimeh, T. Ergin, J. Mueller, N. Stenger, and M. Wegener, “Photorealistic images of carpet cloaks,” Opt. Express 17(22), 19328–19336 (2009). [CrossRef] [PubMed]
  29. 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]
  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] [PubMed]
  31. B. Zhang, Y. Luo, X. Liu, and G. Barbastathis, “Macroscopic invisibility cloak for visible light,” Phys. Rev. Lett. 106(3), 033901 (2011). [CrossRef] [PubMed]
  32. J. Fischer, T. Ergin, and M. Wegener, “Three-dimensional polarization-independent visible-frequency carpet invisibility cloak,” Opt. Lett. 36(11), 2059–2061 (2011). [CrossRef] [PubMed]
  33. Y. A. Urzhumov and D. R. Smith, “Transformation optics with photonic band gap media,” Phys. Rev. Lett. 105(16), 163901 (2010). [CrossRef] [PubMed]
  34. C. Luo, S. Johnson, J. Joannopoulos, and J. Pendry, “All-angle negative refraction without negative effective index,” Phys. Rev. B 65(20), 201104 (2002). [CrossRef]
  35. V. Mocella, S. Cabrini, A. S. P. Chang, P. Dardano, L. Moretti, I. Rendina, D. Olynick, B. Harteneck, and S. Dhuey, “Self-collimation of light over millimeter-scale distance in a quasi-zero-average-index metamaterial,” Phys. Rev. Lett. 102(13), 133902 (2009). [CrossRef] [PubMed]
  36. S. Kocaman, R. Chatterjee, N. C. Panoiu, J. F. McMillan, M. B. Yu, R. M. Osgood, D. L. Kwong, and C. W. Wong, “Observation of zeroth-order band gaps in negative-refraction photonic crystal superlattices at near-infrared frequencies,” Phys. Rev. Lett. 102(20), 203905 (2009). [CrossRef] [PubMed]
  37. X. Huang, Y. Lai, Z. H. Hang, H. Zheng, and C. T. Chan, “Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials,” Nat. Mater. 10(8), 582–586 (2011). [CrossRef] [PubMed]
  38. J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton University Press, 2008).

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

Figures

Fig. 1 Fig. 2 Fig. 3
 
Fig. 4
 

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited