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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 20 — Sep. 27, 2010
  • pp: 21419–21426
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A chirality switching device designed with transformation optics

Yuan Shen, Kun Ding, Wujiong Sun, and Lei Zhou  »View Author Affiliations


Optics Express, Vol. 18, Issue 20, pp. 21419-21426 (2010)
http://dx.doi.org/10.1364/OE.18.021419


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Abstract

Based on transformation optics theory, we designed a chirality switching device, such that an object hidden inside would exhibit a reversed chirality (i.e., from left-handedness to right-handedness) for an observer at the far field. Distinct from a perfect mirror which also creates a chirality-reversed image, our device makes the original object completely invisible to the far field observer. Numerical simulations are employed to demonstrate the functionalities of the designed devices in both two- and three-dimensional spaces.

© 2010 OSA

1. Introduction

Molecular chirality plays a crucial role in biochemistry and asymmetric synthesis [1

1. G. Q. Lin, Y. M. Li, and A. S. C. Chan, Principles and Applications of Asymmetric Synthesis, (John Wiley & Sons, New York, 2001), Chap. 1.

]. For example, while L-form amino acids are considered as the building blocks of most natural proteins, their mirror images find no way inside life on earth [2

2. U. J. Meierhenrich, Amino Acids and the Asymmetry of Life, (Springer, Berlin, 2008), Chap. 1.

]. It is thus highly desirable if the chirality of a certain object can be freely “tuned”. Unfortunately, this is not an easy task since one may have to change the object’s structure mechanically. Here, we demonstrate an optical approach to “change” the chirality of an object. We show that an optical device can be designed, such that any object (with certain chirality) placed inside the device would exhibit a reversed chirality for an observer at the far field.

One may argue that a perfect mirror can achieve this end since it creates a virtual image with opposite chirality. However, in this approach, the original object itself is still visible to the observer, and the same is true for the mirror. Material with refractive index (n) equal to −1 behaves like a mirror plane on its interface with air [3

3. J. B. Pendry and S. A. Ramakrishna, “Focussing Light Using Negative Refraction,” J. Phys. Condens. Matter 15(37), 6345–6364 (2003). [CrossRef]

]. As shown in Fig. 1(a)
Fig. 1 (a) A point source placed inside a semi-infinite n = −1 medium creates a real image in the air region. (b) A point source placed inside a finite-sized n = −1 medium creates multiple images, each observable within only a certain viewing angle range.
, light rays emitted from a source [4

4. In this paper, we consider only those sources emitting light rays symmetrically, i.e., for any ray emitted from the source, we can always find another emitted ray going to the opposite direction.

] located inside an n = −1 semi-infinite medium are focused to a point in air region [5

5. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000). [CrossRef] [PubMed]

]. Therefore, for an observer at the far field (in the air region side), the realistic point source seems to be moved from its original position (inside the n = −1 medium) to its symmetry point [6

6. D. Maystre and S. Enoch, “Perfect lenses made with left-handed materials: Alice’s mirror?” J. Opt. Soc. Am. A 21(1), 122–131 (2004). [CrossRef]

]. Push the idea forward, a chiral object (assumed as containing many point sources) placed inside the n = −1 medium must exhibit a reversed chirality for a far-field observer at the air side. Unfortunately, the device shown in Fig. 1(a) can never be realized since it requires an infinite boundary. One may immediately think of a bounded area of n = −1 medium, as shown in Fig. 1(b). However, such a device would create multiple images, each observable only within a certain range of viewing angles, as schematically depicted in Fig. 1(b). Recently, illusion optics theory [7

7. Y. Lai, J. Ng, H. Y. Chen, D. Z. Han, J. J. Xiao, Z. Q. Zhang, and C. T. Chan, “Illusion optics: the optical transformation of an object into another object,” Phys. Rev. Lett. 102(25), 253902 (2009). [CrossRef] [PubMed]

] was established, which could in principle create any optical illusion as desired. The idea is to first optically cancel an object with complementary medium [8

8. Y. Lai, H. Y. 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]

] and then project an illusion of different object through some restoring medium [7

7. Y. Lai, J. Ng, H. Y. Chen, D. Z. Han, J. J. Xiao, Z. Q. Zhang, and C. T. Chan, “Illusion optics: the optical transformation of an object into another object,” Phys. Rev. Lett. 102(25), 253902 (2009). [CrossRef] [PubMed]

]. While in principle one can employ this theory to design a chirality-switching device, such an approach suffers a limitation that the objects hidden inside the device must be transparent.

2. Operation, transformation, and device parameters

Before we present our design, it is helpful to briefly review the essence of the transformation optics theory [9

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

]. The theory actually established a correspondence between the material filling the space (the transformation medium) and a coordinate transformation [10

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

], which is mediated by a particular operation performed in certain space region [11

11. L. V. Ahlfors, Complex Analysis, (McGraw-Hill, New York, 1979), Chap. 3&8.

]. For instance, the operation adopted to design the field rotator [12

12. H. Y. Chen and C. T. Chan, “Transformation Media that Rotate Electromagnetic Fields,” Appl. Phys. Lett. 90(24), 241105 (2007). [CrossRef]

] is a proper rotation of 3D coordinate system by a specific angle; a field concentrator [13

13. M. Rahm, D. Schurig, D. A. Robertsa, S. A. Cummera, D. R. Smith, and J. B. Pendry, “Design of Electromagnetic Cloaks and Concentrators Using Form-invariant Coordinate Transformations of Maxwell’s Equations,” Phot. Nano.: Fund. Appl. 6(1), 87–95 (2008). [CrossRef]

] could be viewed as a plate employing an stretching operation; and one can easily identify the operations for the hyper-lens (stretching) [14

14. Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Optical Hyperlens: Far-field Imaging Beyond the Diffraction Limit,” Opt. Express 14(18), 8247–8256 (2006). [CrossRef] [PubMed]

] and the beam shifter (twisting) [15

15. 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]

]. The perfect lens effect [5

5. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000). [CrossRef] [PubMed]

] can also be explained by the transformation optics theory, and the operation was identified by Leonhardt and Philbin as a space “folding” [16

16. U. Leonhardt and T. G. Philbin, “General Relativity in Electrical Engineering,” N. J. Phys. 8(10), 247 (2006). [CrossRef]

]. The folded area in electromagnetic (EM) space turns out to be a slab of material with negative n in the physical space [10

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

]. Therefore, to design any functional device based on transformation optics, the first and most important task is to identify the “operation” required to achieve the desired function. In addition, to make a good design, the operation should follow some general rules. First, the operation should be performed uniformly if one wants to obtain a homogeneous transformation medium [10

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

]. Second, the operation cannot tear any boundary if one does not want reflections at the boundaries of the transformation medium [17

17. L. Bergamin, “Electromagnetic Fields and Boundary Conditions at the Interface of Generalized Transformation Media,” Phys. Rev. A 80(6), 063835 (2009). [CrossRef]

].

Transformation optics theory [9

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

] tells us that an empty space with a distorted geometry is equivalent to a non-distorted space but filled with some material called transformation medium. According to the operation defined in Fig. 2(a-d), we can easily get the coordinate transformation and material properties of the transformation medium based on the theory developed in [9

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

]. Let {x,y,z} and {u,v,w} be the coordinate systems in physical and EM space, respectively, we summarized the coordinate transformations and the corresponding materials parameters in different regions [see definitions in Fig. 3(a)
Fig. 3 (a) Radiation pattern of a line source placed inside the 2D device. (b) Radiation pattern of a line source placed in free space, at the mirror-reflection position of the source. (c) Radiation pattern of a directional source inside the device, which emits rays only along two directionsk=±(122,122,0)k0. (d) Radiation pattern of another direction source placed in free space, which emits rays only along two directionsk=±(122,122,0)k0.
] in Table 1

Table 1. Coordinate transformation and material parameters in different space regions

table-icon
View This Table
.

At this point, it is intuitive to compare the proposed device with the scheme that we discussed in Fig. 1(b). In fact, the core area of the proposed device is nothing but an n = −1 medium in a rhombus shape (see property of region V in Table 1). However, we have mentioned that such a system alone does not behave as a perfect mirror, as schematically shown in Fig. 1(b). The recipe here is to add four pieces of transformation media (regions I-IV) to surround such a rhombus n = −1 area, which can help resolve the problems existing for a rhombus n = −1 device alone. It is worth mentioning that, although εr,μr look quite complicated at first sight, they can be easily diagonalized so that the transformation media are not that difficult to realize practically.

3. Numerical verifications

Full wave numerical simulations [18

18. COMSOL Multi-physics 3.5, developed by COMSOL ©, network license (2008).

] are performed to demonstrate the functionality of the proposed device. We consider the simple 2D case first, and then extend to more complicated 3D case. However, an important difference between 2D and 3D cases should be mentioned first. Since a 2D structure does not exhibit real chirality, the 2D device works actually as a transforming mirror for 2D objects precisely speaking. On the other hand, for a realistic 3D chiral structure, the 3D device can indeed optically switch its chirality as a perfect mirror. In the following, we demonstrate two important characteristics for the proposed device:

  • 1) for any true light source, the device effectively relocate its position to the mirror-reflection symmetry point with respect to the central plane of the device, for an observer at the far field;
  • 2) for a ray (radiated from a true source) which originally propagates to left (right) direction, the device effectively redirect its propagation to right (left) direction, again for an observer at the far field.

We now demonstrate the first characteristics. In our simulations, we set a = 6.25 [19

19. All lengths are rescaled by the working wavelengths taken in the simulations.

] and assumed that the device medium properties were given by Table 1, and then placed a line source at a point (x = 1.25, y = 1.25, z = 0) inside the device. The calculated radiation pattern (only Ez component is shown) of this source is depicted in Fig. 3(a). While the field distribution inside the device is rather complex, we find surprisingly that the radiation pattern outside the device is quite regular. As a comparison, we show in Fig. 3(b) the radiation pattern of a line source placed at a point (x = −1.25, y = 1.25, z = 0) with the device removed [20

20. Here the shape of device is shown only for identifying the position.

]. Comparison between Fig. 3(a) and Fig. 3(b) show that they are essentially the same, demonstrating that the proposed device can indeed effectively relocate the position of a source to its mirror-reflection symmetry point.

Now extend the discussions to an arbitrary 2D source placed inside the device. Suppose we could regard the complex-shaped source as a collection of line sources, then applying the above two characteristics, we understand that our device effectively replaces the original source by a virtual source, which is the mirror image of the original one, for a far field observer.

One important merit of our device is that it makes itself and the original object invisible to a far field observer, completely different from an ordinary mirror. This has already been partially demonstrated in Fig. 3(a) and (c), where only the virtual mirror-image source can be detected for an observer at the far field. More interestingly, our device is also invisible to external radiations. To demonstrate this effect, we depicted in Fig. 4
Fig. 4 Propagations of plane EM waves polarized with E||z^ passing through the device along (a) x^direction and (b) (x^+y^)direction.
the simulated field distributions when the device is under radiations of external plane waves polarized withE||z^. While EM waves are distorted within the device, they are restored to their original propagation direction after leaving the device, making the device completely undetectable for a far field observer. This is easy to understand from the operation as specified in Fig. 2(a-d), since the operation is performed with a restricted area and thus does not affect the light ray trajectories outside the device. In particular, we note that the propagation directions inside the inner rhombus (with n = −1) are different from the original direction, in consistency with Fig. 2(d). Another merit of our device is that it does not require the objects hidden inside to be transparent as in illusion optics theory [7

7. Y. Lai, J. Ng, H. Y. Chen, D. Z. Han, J. J. Xiao, Z. Q. Zhang, and C. T. Chan, “Illusion optics: the optical transformation of an object into another object,” Phys. Rev. Lett. 102(25), 253902 (2009). [CrossRef] [PubMed]

], and therefore, our device may have a wide application range.

The extension to the 3D case is straightforward. When taking the z coordinate into consideration, the design for 2D space extends to infinity along the z axis. However, a real 3D device must be finite along all directions. Unfortunately, just truncating the device at some points at the z axis does not work well since the upper and lower boundaries may cause undesirable reflections. This problem can be solved by taking advantage of the fact that the functionality of the 2D device is independent of the value of a (see Table 1). Therefore, shrinking the cross section of the device gradually to zero as |z| increases to a certain value, we obtain a design for the 3D device which is finite along all directions. By doing so, all potential reflections could be eliminated since the device no longer tears up any boundary. Inset to Fig. 5(a)
Fig. 5 (a) Radiation pattern of a point source placed inside the 3D chirality-switching device in an octahedron shape (see inset). For the purpose of a better illustration, the coordinate system is readjusted with the coordinate origin shifted to the virtual source position. We plot the x, y and z components of electric field in the yz, zx and xy planes of the new coordinate system, correspondingly. Edges of the device are indicated by solid lines in the figure. (b) Electric field distribution on the xy plane with z = −0.235, with the crossing point of the two dotted lines representing the origin of those circular wave-fronts. Fields in some regions are out of scale.
shows the designed 3D chirality-switching device, which is an octahedron with its material parameters obtained by substituting a in Table 1 by
a(z)=a0|z|
(1)
for the xy plane at a particular z point. Obviously, the total height of the 3D device is 2a 0.

Numerical simulations [18

18. COMSOL Multi-physics 3.5, developed by COMSOL ©, network license (2008).

] are again performed to demonstrate the chirality switching effect for this 3D device. Restricted by our computational resources, we cannot study a real chiral object placed inside the device. Instead, we consider only a point source and demonstrate that the device behaves as a perfect mirror for this point source. The size of the device is set as a 0 = 2.35 [19

19. All lengths are rescaled by the working wavelengths taken in the simulations.

]. As shown in Fig. 5(a), we placed a point source [22

22. The point source consists of three mutually orthogonal antennae, with pumping fields carefully adjusted to ensure that the resultant radiation pattern in free space exhibits a spherical symmetry.

] at point (x = −0.235, y = 0.235, z = −0.235), which is at a distance of 0.235 away from the middle symmetry plane, and then performed numerical simulations to compute the radiation patterns. While the near field distribution is very complicated, Fig. 5(a) shows that the far-field wave pattern is rather regular, and it seems that the waves are coming from a virtual point source. To identify the position of this virtual source, we drew in Fig. 5(b) the 2D near field pattern on the xy plane with z = −0.235 (the plane where the source is located). We found that this virtual source is located at point (x = 0.235, y = 0.235, z = −0.235), which is at the exact mirror-reflection symmetry point of the true source position. This fact has clearly demonstrated that the device can indeed work as a mirror for any source placed inside, while keeping the device and the original object invisible to an observer located at the far field. Extending the discussions to a more complicated 3D source, we expect that a realistic chiral object hidden inside such a device would exhibit an opposite chirality for an observer at the far field.

4. Conclusions

In conclusion, we have identified a new type of operation performed on a fixed area in EM space, based on which a device can be designed which can optically change the chirality of an object hidden inside. The functionalities of the proposed device are demonstrated in both 2D and 3D spaces by full wave numerical simulations.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (60725417, 60990321) and Shanghai science and technology committee (No. 08dj1400302).

References and links

1.

G. Q. Lin, Y. M. Li, and A. S. C. Chan, Principles and Applications of Asymmetric Synthesis, (John Wiley & Sons, New York, 2001), Chap. 1.

2.

U. J. Meierhenrich, Amino Acids and the Asymmetry of Life, (Springer, Berlin, 2008), Chap. 1.

3.

J. B. Pendry and S. A. Ramakrishna, “Focussing Light Using Negative Refraction,” J. Phys. Condens. Matter 15(37), 6345–6364 (2003). [CrossRef]

4.

In this paper, we consider only those sources emitting light rays symmetrically, i.e., for any ray emitted from the source, we can always find another emitted ray going to the opposite direction.

5.

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000). [CrossRef] [PubMed]

6.

D. Maystre and S. Enoch, “Perfect lenses made with left-handed materials: Alice’s mirror?” J. Opt. Soc. Am. A 21(1), 122–131 (2004). [CrossRef]

7.

Y. Lai, J. Ng, H. Y. Chen, D. Z. Han, J. J. Xiao, Z. Q. Zhang, and C. T. Chan, “Illusion optics: the optical transformation of an object into another object,” Phys. Rev. Lett. 102(25), 253902 (2009). [CrossRef] [PubMed]

8.

Y. Lai, H. Y. 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]

9.

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

10.

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

11.

L. V. Ahlfors, Complex Analysis, (McGraw-Hill, New York, 1979), Chap. 3&8.

12.

H. Y. Chen and C. T. Chan, “Transformation Media that Rotate Electromagnetic Fields,” Appl. Phys. Lett. 90(24), 241105 (2007). [CrossRef]

13.

M. Rahm, D. Schurig, D. A. Robertsa, S. A. Cummera, D. R. Smith, and J. B. Pendry, “Design of Electromagnetic Cloaks and Concentrators Using Form-invariant Coordinate Transformations of Maxwell’s Equations,” Phot. Nano.: Fund. Appl. 6(1), 87–95 (2008). [CrossRef]

14.

Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Optical Hyperlens: Far-field Imaging Beyond the Diffraction Limit,” Opt. Express 14(18), 8247–8256 (2006). [CrossRef] [PubMed]

15.

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]

16.

U. Leonhardt and T. G. Philbin, “General Relativity in Electrical Engineering,” N. J. Phys. 8(10), 247 (2006). [CrossRef]

17.

L. Bergamin, “Electromagnetic Fields and Boundary Conditions at the Interface of Generalized Transformation Media,” Phys. Rev. A 80(6), 063835 (2009). [CrossRef]

18.

COMSOL Multi-physics 3.5, developed by COMSOL ©, network license (2008).

19.

All lengths are rescaled by the working wavelengths taken in the simulations.

20.

Here the shape of device is shown only for identifying the position.

21.

The directional source is formed by an array of 6 × 6 line sources (with lattice constant 0.5.). The pumping fields of these sources are chosen as (+11+11+11+12+22+21+12+44+21+12+44+21+12+22+21+11+11+11)

22.

The point source consists of three mutually orthogonal antennae, with pumping fields carefully adjusted to ensure that the resultant radiation pattern in free space exhibits a spherical symmetry.

OCIS Codes
(230.0230) Optical devices : Optical devices
(260.2110) Physical optics : Electromagnetic optics
(160.3918) Materials : Metamaterials

ToC Category:
Physical Optics

History
Original Manuscript: August 5, 2010
Revised Manuscript: September 2, 2010
Manuscript Accepted: September 16, 2010
Published: September 23, 2010

Citation
Yuan Shen, Kun Ding, Wujiong Sun, and Lei Zhou, "A chirality switching device designed with transformation optics," Opt. Express 18, 21419-21426 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-20-21419


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References

  1. G. Q. Lin, Y. M. Li, and A. S. C. Chan, Principles and Applications of Asymmetric Synthesis, (John Wiley & Sons, New York, 2001), Chap. 1.
  2. U. J. Meierhenrich, Amino Acids and the Asymmetry of Life, (Springer, Berlin, 2008), Chap. 1.
  3. J. B. Pendry and S. A. Ramakrishna, “Focussing Light Using Negative Refraction,” J. Phys. Condens. Matter 15(37), 6345–6364 (2003). [CrossRef]
  4. In this paper, we consider only those sources emitting light rays symmetrically, i.e., for any ray emitted from the source, we can always find another emitted ray going to the opposite direction.
  5. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000). [CrossRef] [PubMed]
  6. D. Maystre and S. Enoch, “Perfect lenses made with left-handed materials: Alice’s mirror?” J. Opt. Soc. Am. A 21(1), 122–131 (2004). [CrossRef]
  7. Y. Lai, J. Ng, H. Y. Chen, D. Z. Han, J. J. Xiao, Z. Q. Zhang, and C. T. Chan, “Illusion optics: the optical transformation of an object into another object,” Phys. Rev. Lett. 102(25), 253902 (2009). [CrossRef] [PubMed]
  8. Y. Lai, H. Y. 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]
  9. U. Leonhardt, “Optical Conformal Mapping,” Science 312(5781), 1777–1780 (2006). [CrossRef] [PubMed]
  10. H. Chen, C. T. Chan, and P. Sheng, “Transformation Optics and Metamaterials,” Nat. Mater. 9(5), 387–396 (2010). [CrossRef] [PubMed]
  11. L. V. Ahlfors, Complex Analysis, (McGraw-Hill, New York, 1979), Chap. 3&8.
  12. H. Y. Chen and C. T. Chan, “Transformation Media that Rotate Electromagnetic Fields,” Appl. Phys. Lett. 90(24), 241105 (2007). [CrossRef]
  13. M. Rahm, D. Schurig, D. A. Robertsa, S. A. Cummera, D. R. Smith, and J. B. Pendry, “Design of Electromagnetic Cloaks and Concentrators Using Form-invariant Coordinate Transformations of Maxwell’s Equations,” Phot. Nano.: Fund. Appl. 6(1), 87–95 (2008). [CrossRef]
  14. Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Optical Hyperlens: Far-field Imaging Beyond the Diffraction Limit,” Opt. Express 14(18), 8247–8256 (2006). [CrossRef] [PubMed]
  15. 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]
  16. U. Leonhardt and T. G. Philbin, “General Relativity in Electrical Engineering,” N. J. Phys. 8(10), 247 (2006). [CrossRef]
  17. L. Bergamin, “Electromagnetic Fields and Boundary Conditions at the Interface of Generalized Transformation Media,” Phys. Rev. A 80(6), 063835 (2009). [CrossRef]
  18. COMSOL Multi-physics 3.5, developed by COMSOL ©, network license (2008).
  19. All lengths are rescaled by the working wavelengths taken in the simulations.
  20. Here the shape of device is shown only for identifying the position.
  21. The directional source is formed by an array of 6 × 6 line sources (with lattice constant 0.5.). The pumping fields of these sources are chosen as 
(+1−1+1−1+1−1+1−2+2−2+2−1+1−2+4−4+2−1+1−2+4−4+2−1+1−2+2−2+2−1+1−1+1−1+1−1)
  22. The point source consists of three mutually orthogonal antennae, with pumping fields carefully adjusted to ensure that the resultant radiation pattern in free space exhibits a spherical symmetry.

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