## Newtonian photorealistic ray tracing of grating cloaks and correlation-function-based cloaking-quality assessment |

Optics Express, Vol. 19, Issue 7, pp. 6078-6092 (2011)

http://dx.doi.org/10.1364/OE.19.006078

Acrobat PDF (2105 KB)

### Abstract

Grating cloaks are a variation of dielectric carpet (or ground-plane) cloaks. The latter were introduced by Li and Pendry. In contrast to the numerical work involved in the quasi-conformal carpet cloak, the refractive-index profile of a conformal grating cloak follows a closed and exact analytical form. We have previously mentioned that finite-size conformal grating cloaks may exhibit better cloaking than usual finite-size carpet cloaks. In this paper, we directly visualize their performance using photorealistic ray-tracing simulations. We employ a Newtonian approach that is advantageous compared to conventional ray tracing based on Snell’s law. Furthermore, we quantify the achieved cloaking quality by computing the cross-correlations of rendered images. The cross-correlations for the grating cloak are much closer to 100% (*i.e.*, ideal) than those for the Gaussian carpet cloak.

© 2011 OSA

## 1. Introduction

*selected*rays in two dimensions (as done in our Ref [15

15. R. Schmied, J. C. Halimeh, and M. Wegener, “Conformal carpet and grating cloaks,” Opt. Express **18**(23), 24361–24367 (2010). [CrossRef] [PubMed]

18. A. J. Danner, “Visualizing invisibility: metamaterials-based optical devices in natural environments,” Opt. Express **18**(4), 3332–3337 (2010). [CrossRef] [PubMed]

## 2. The grating cloak

*n*of the grating cloak has been given [15

15. R. Schmied, J. C. Halimeh, and M. Wegener, “Conformal carpet and grating cloaks,” Opt. Express **18**(23), 24361–24367 (2010). [CrossRef] [PubMed]

*u*,

*v*) in Eq. (10) in Ref [15

15. R. Schmied, J. C. Halimeh, and M. Wegener, “Conformal carpet and grating cloaks,” Opt. Express **18**(23), 24361–24367 (2010). [CrossRef] [PubMed]

*x*=

*u*,

*y*=

*v*,

*z*) in three dimensions, we haveHere,

*W*

_{0}is the Lambert function (or product logarithm). The constant wave number

*k*is given by

*k*= 2

*π*/Λ with the real-space grating period Λ; the complex-valued constant

*c*is given by

*c*= i

*A*with real-valued

*A*, where 2

*A*is the peak-to-peak vertical grating amplitude (see Eq. (2)). Thus, numerical errors in determining the index profile (as,

*e.g.*, possible for the quasi-conformal carpet cloak requiring numerical minimization of the modified-Liao functional) do not play any role at all in our present work. We do not consider any frequency dependence of

*n*.

*y*(

*x*) of the metal grating off of which light rays are reflected and which is to be cloaked by the refractive-index profile given by Eq. (1) has the shape of a trochoid given by the parametric form [15

**18**(23), 24361–24367 (2010). [CrossRef] [PubMed]

*ξ*(corresponding to

*x*in Ref [15

**18**(23), 24361–24367 (2010). [CrossRef] [PubMed]

*z*-direction and can be brought into the explicit formwith integer

*N*= -∞, …, + ∞ for the different grating maxima. The “-“ sign in Eq. (3) applies for the left-hand-side slope of each maximum, the “+” sign for the right-hand-side slope. In the shallow-grating limit defined by

*2π A*/Λ <<1, hence

*x ≈ξ*in Eq. (2), one gets the approximate explicit formHowever, for the numerical evaluation of the ray trajectories we will rather use the exact forms of Eqs. (2) and (3) throughout this work. Figure 1(a) shows an example for the normalized grating-cloak model parameters Λ = 1 and

*A*= 3/(16

*π*)≈0.06 that lead to experimentally accessible refractive indices. In Figs. 3 -7 , one normalized unit corresponds to an actual 19 cm.

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

**18**(23), 24361–24367 (2010). [CrossRef] [PubMed]

*h*and the width

*w*. However, for the actual calculations, we use the exact implicit form given in Ref [15

**18**(23), 24361–24367 (2010). [CrossRef] [PubMed]

*w =*0.306 and

*h*= 0.13.

## 3. Newtonian photorealistic ray tracing

14. 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]

19. T. Ergin, J. C. Halimeh, N. Stenger, and M. Wegener, “Optical microscopy of 3D carpet cloaks:ray-tracing calculations,” Opt. Express **18**(19), 20535–20545 (2010). [CrossRef] [PubMed]

18. A. J. Danner, “Visualizing invisibility: metamaterials-based optical devices in natural environments,” Opt. Express **18**(4), 3332–3337 (2010). [CrossRef] [PubMed]

29. K. Niu, C. Song, and M.-L. Ge, “The geodesic form of light-ray trace in the inhomogeneous media,” Opt. Express **17**(14), 11753–11767 (2009). [CrossRef] [PubMed]

*r*and the components of the velocity vector

_{i}*v*Identifying the first term as the force component

_{i}*F*, the second term as the temporal derivative of the momentum component, and additionally assuming a constant particle mass

_{i}*m*, one immediately arrives at Newton’s second law for the accelerationWe can now simply repeat this procedure for light rays, where Hamilton’s principle (Eq. (6)) has to be replaced by Fermat’s principle,

*i.e.*, byIn analogy to Eq. (7), this leads toand upon inserting the modulus of the local phase velocity of light

*v*=

*c*

_{0}/

*n*, which is known

*a priori*at each point in space, to the Newtonian equation of motionThe effective light-ray acceleration on the right-hand side of Eq. (11) vanishes in homogeneous media and thus outside the cloak. Inside the cloak, we solve this ordinary second-order differential equation (Eq. (11)) numerically for each ray. At the interface between the cloak and the surrounding air (or embedding medium), Snell’s law and the Fresnel equations correctly describe the refraction and reflection of the ray due to the discontinuity in the refractive index. We neglect rays which are Fresnel reflected more than once, because their intensity will be negligible in practice.

*W*

_{0}(see Eq. (1)) Equation (12) can be inserted on the right-hand side of Eq. (11). We mention that, even though the refractive-index profiles used in this work are translationally-invariant in the

*z*-direction, and thus the

*z*-component of the gradient of

*n*is zero throughout space, the ray acceleration in the

*z*-direction in Eq. (11) does in general not vanish.

## 4. Rendered images

*focal*field of view (

*i.e.*, full opening angles of 50° horizontally and 42° vertically) is located between her eyes, which are centered in the photograph in Fig. 2(a). The distance between model and mirror is

*d*= 0.5 m (or 2.64 normalized units) – a typical distance for standing in front of your bathroom mirror. The model is 1.7 m (or 9.0 normalized units) tall. In the coordinate system shown in Figs. 1(a) and 1(b), the model is located at the top and looks downward onto the (a) grating or (b) Gaussian bump. For case (a), she sees three grating maxima within her vertical field of view (FOV): one at the center and two at the edges. For case (b), the single maximum is at the center of her FOV.

*i.e.*, tan(42°/2) = Λ/

*d*). Between maxima and minima, inflection points of the mirror profile occur. At each inflection point, the image turns from right-side-up to upside-down or

*vice versa*. The combination of these aspects leads to highly complex and pronounced distortions with respect to the reference, Fig. 3(a).

*i.e.*, all the distortions in the respective panels (b) should be compensated for. This is indeed very nearly the case on the vertical line through the center of the images, but increasing deviations occur towards the sides. The grating cloak in Fig. 3 appears to perform better than the Gaussian conformal map in Fig. 4. We will quantify these aspects by evaluating cross-correlation coefficients of panels (a) and (c) in the following section.

*i.e.*, the index of air is taken as unity and the indices of the cloaks do exhibit values below unity, with the minimum values given in Fig. 1. This aspect makes experimental realization difficult. If one divides all refractive indices by the minimum value

*n*

_{min}, the results remain unchanged. However, it is more realistic to assume that we keep an air refractive index of

*n*= 1 (rather than

*n*= 1/

*n*

_{min}>1). It has been pointed out repeatedly [6

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

14. 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]

*i.e.*, 1.1 normalized units or 20.8 cm), in Fig. 6 (smaller cloak height of 0.55 normalized units or 10.4 cm), and in Fig. 7 (yet smaller cloak height of 0.41 normalized units or 7.8 cm). The “ideal” images as depicted in panels (a) of Figs. 5-7 are equivalent to adding a homogeneous dielectric plate with refractive index

*n*= 1/

*n*

_{min}>1 and with the same height as the respective cloak onto the flat mirror. These images are all different and

*not*identical to Fig. 3(a) because the homogeneous dielectric plate with a refractive index larger than that of air effectively reduces the field of view. This effect can be seen by comparing the books at the upper right-hand side corner of Fig. 3(a) and Fig. 5(a). It decreases with decreasing dielectric-plate thickness from Fig. 5(a) via Fig. 6(a) to Fig. 7(a). Furthermore, one can see very faint Fresnel reflections in the high-resolution versions of panels (a) of Figs. 5-7 (see,

*e.g.*, necklace on light blue shirt in Fig. 5(a)). The Fresnel transmissions also make the primary images very slightly dimmer than the reference image in Fig. 3(a).

*n*

_{min}far away from the corrugation as discussed in Ref [15

**18**(23), 24361–24367 (2010). [CrossRef] [PubMed]

## 5. Correlation function as quantitative measure of cloaking quality

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

**18**(23), 24361–24367 (2010). [CrossRef] [PubMed]

30. M. James, *Pattern Recognition* (John Wiley & Sons, 1988). [PubMed]

31. J. L. Rodgers and W. A. Nicewander, “Thirteen ways to look at the correlation coefficient,” Am. Stat. **42**(1), 59–66 (1988). [CrossRef]

*f*(

*x,y*)≥0 and

*g*(

*x*,

*y*)≥0 represent the intensity or gray levels of two monochrome two-dimensional finite-size photographs (like,

*e.g.*, the images in Figs. 3-7) on an arbitrary scale, where

*x*and

*y*are the orthogonal coordinates in the photographic plane. (More generally, you can consider three such functions for the three principal colors in color photographs.) For example, let

*f*(

*x*,

*y*) be the unperturbed or reference image (

*i.e.*, no object and no cloak) and

*g*(

*x*,

*y*) be the image taken with object and cloak. Thus, we will assume that the spatial average value of

*g*is not larger than the spatial average value of

*f*. First, we subtract their respective spatial average valuesNext, we define a two-dimensional image cross-correlation function,

*C*, according to

*i.e.*, for

*f*(

*x*,

*y*) =

*g*(

*x*,

*y*) ∀ (

*x*,

*y*), we obtain the autocorrelation

*C*(0,0) = + 100%. The value of

*C*will decay away from the maximum at (Δ

*x*= 0,Δ

*y*= 0) towards 0% depending on the special properties of the photographic image

*f*(

*x*,

*y*). For example, periodic patterns in

*f*(

*x*,

*y*) will give rise to multiple maxima of

*C*(Δ

*x*,Δ

*y*). Non-periodic images

*f*(

*x*,

*y*) generally only lead to a single pronounced global maximum of

*C*(Δ

*x*,Δ

*y*) at (Δ

*x =*0,Δ

*y =*0). If the image

*g*(

*x*,

*y*) =

*a*×

*f*(

*x,y*) with factor 0≤

*a*≤1 is just dimmer than the reference image

*f*(

*x,y*),

*e.g.*, due to some absorption or scattering of the cloaking device, we get

*C*(0,0) =

*a*≤100%. If

*g*(

*x*,

*y*) is the “inverse” or complement (

*i.e.*, black replaced by white and

*vice versa*) of

*f*(

*x*,

*y*), we get

*C*(0,0) = –100%. If the image

*g*(

*x*,

*y*) is identical to the original

*f*(

*x,y*), but shifted within the photographic

*xy*-plane, a correlation maximum with a value approaching 100% (or strictly equal to + 100% for infinitely large photographs) appears at some shifted position (Δ

*x≠*0,Δ

*y≠*0). If

*g*(

*x*,

*y*) has nothing to do with

*f*(

*x*,

*y*), the correlation function will exhibit values near 0% for all (Δ

*x*, Δ

*y*).

*C*(0,0) or the peak value of

*C*(Δ

*x≠*0,Δ

*y≠*0) as the requested single-number measure for cloaking quality. The former is stringent, the latter is more generous with respect to possible shifts in the images (like,

*e.g.*, for the beam displacement of the carpet cloak). This shift could also be quoted.

*Note that either of the two measures based on*

*Eq. (15)*

*takes advantage of the entire images and is not based on individual points of the images only.*

*f*(

_{i}*x,y*) and corresponding cloaked images

*g*(

_{i}*x*,

*y*) with index

*i*to eliminate any dependence on the particular image scenery.

*i.e.*, to a vertical line through the center of the images in Figs. 3-7. Note that the grating cloak (like the carpet cloak) is exact in this two-dimensional (2D) world only. Indeed, under these conditions, the correlation is

*C*(0,0) = −20% without cloak and as large as 99% with cloak – essentially perfect cloaking is achieved. Going to a three-dimensional setting reduces the value for the cloaked case to 91% for 10° FOV, to 73% for 25° FOV and to a mere 51% for 50° FOV; whereas, the values for the uncloaked case are all very small or even negative, indicating strongly distorted images.

13. B. Zhang, T. Chan, and B.-I. Wu, “Lateral shift makes a ground-plane cloak detectable,” Phys. Rev. Lett. **104**(23), 233903 (2010). [CrossRef] [PubMed]

**18**(23), 24361–24367 (2010). [CrossRef] [PubMed]

**18**(23), 24361–24367 (2010). [CrossRef] [PubMed]

*C*further decreases.

## 6. Conclusion

*quantified*by the cross-correlation coefficient under realistic conditions. We emphasize that the parameter choices in our work correspond to locally isotropic refractive indices between a minimum of 1.0 and a maximum of 2.2. These values are accessible throughout the entire visible spectral region with little dispersion,

*e.g.*, by nanostructuring titania along the lines of Ref [11

11. 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]

## Acknowledgements

## References and links

1. | J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science |

2. | U. Leonhardt, “Optical conformal mapping,” Science |

3. | V. M. Shalaev, “Physics. Transforming light,” Science |

4. | H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Mater. |

5. | M. Wegener and S. Linden, “Shaping optical space with metamaterials,” Phys. Today |

6. | J. Li and J. B. Pendry, “Hiding under the carpet: a new strategy for cloaking,” Phys. Rev. Lett. |

7. | R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science |

8. | J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. |

9. | L. H. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nat. Photonics |

10. | 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 |

11. | T. Ergin, N. Stenger, P. Brenner, J. B. Pendry, and M. Wegener, “Three-dimensional invisibility cloak at optical wavelengths,” Science |

12. | H. F. Ma and T. J. Cui, “Three-dimensional broadband ground-plane cloak made of metamaterials,” Nat. Commun. |

13. | B. Zhang, T. Chan, and B.-I. Wu, “Lateral shift makes a ground-plane cloak detectable,” Phys. Rev. Lett. |

14. | J. C. Halimeh, T. Ergin, J. Mueller, N. Stenger, and M. Wegener, “Photorealistic images of carpet cloaks,” Opt. Express |

15. | R. Schmied, J. C. Halimeh, and M. Wegener, “Conformal carpet and grating cloaks,” Opt. Express |

16. | A. S. Glassner, |

17. | G. Dolling, M. Wegener, S. Linden, and C. Hormann, “Photorealistic images of objects in effective negative-index materials,” Opt. Express |

18. | A. J. Danner, “Visualizing invisibility: metamaterials-based optical devices in natural environments,” Opt. Express |

19. | T. Ergin, J. C. Halimeh, N. Stenger, and M. Wegener, “Optical microscopy of 3D carpet cloaks:ray-tracing calculations,” Opt. Express |

20. | J. L. Synge, |

21. | M. Born, and E. Wolf, |

22. | J. S. Desjardins, “Time-dependent geometrical optics,” J. Opt. Soc. Am. |

23. | J. Molcho and D. Censor, “A simple derivation and an example of Hamiltonian ray propagation,” Am. J. Phys. |

24. | P. S. J. Russell and T. A. Birks, “Hamiltonian optics of nonuniform photonic crystals,” J. Lightwave Technol. |

25. | C. Bellver-Cebreros and M. Rodriguez-Danta, “Eikonal equation from continuum mechanics and analogy between equilibrium of a string and geometrical light rays,” Am. J. Phys. |

26. | Y. Jiao, S. Fan, and D. A. B. Miller, “Designing for beam propagation in periodic and nonperiodic photonic nanostructures: extended Hamiltonian method,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

27. | D. Schurig, J. B. Pendry, and D. R. Smith, “Calculation of material properties and ray tracing in transformation media,” Opt. Express |

28. | D. A. Genov, S. Zhang, and X. Zhang, “Mimicking celestial mechanics in metamaterials,” Nat. Phys. |

29. | K. Niu, C. Song, and M.-L. Ge, “The geodesic form of light-ray trace in the inhomogeneous media,” Opt. Express |

30. | M. James, |

31. | J. L. Rodgers and W. A. Nicewander, “Thirteen ways to look at the correlation coefficient,” Am. Stat. |

**OCIS Codes**

(080.0080) Geometric optics : Geometric optics

(080.2710) Geometric optics : Inhomogeneous optical media

(160.3918) Materials : Metamaterials

(230.3205) Optical devices : Invisibility cloaks

**History**

Original Manuscript: December 23, 2010

Revised Manuscript: February 9, 2011

Manuscript Accepted: February 10, 2011

Published: March 17, 2011

**Citation**

Jad C. Halimeh, Roman Schmied, and Martin Wegener, "Newtonian photorealistic ray tracing of grating cloaks and correlation-function-based cloaking-quality assessment," Opt. Express **19**, 6078-6092 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-7-6078

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### References

- J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006). [CrossRef] [PubMed]
- U. Leonhardt, “Optical conformal mapping,” Science 312(5781), 1777–1780 (2006). [CrossRef] [PubMed]
- V. M. Shalaev, “Physics. Transforming light,” Science 322(5900), 384–386 (2008). [CrossRef] [PubMed]
- H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Mater. 9(5), 387–396 (2010). [CrossRef] [PubMed]
- M. Wegener and S. Linden, “Shaping optical space with metamaterials,” Phys. Today 63(10), 32–36 (2010). [CrossRef]
- J. Li and J. B. Pendry, “Hiding under the carpet: a new strategy for cloaking,” Phys. Rev. Lett. 101(20), 203901 (2008). [CrossRef] [PubMed]
- 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]
- 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]
- 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]
- 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]
- 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]
- H. F. Ma and T. J. Cui, “Three-dimensional broadband ground-plane cloak made of metamaterials,” Nat. Commun. 1(3), 1–6 (2010). [CrossRef]
- B. Zhang, T. Chan, and B.-I. Wu, “Lateral shift makes a ground-plane cloak detectable,” Phys. Rev. Lett. 104(23), 233903 (2010). [CrossRef] [PubMed]
- 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]
- R. Schmied, J. C. Halimeh, and M. Wegener, “Conformal carpet and grating cloaks,” Opt. Express 18(23), 24361–24367 (2010). [CrossRef] [PubMed]
- A. S. Glassner, An Introduction to Ray Tracing (Morgan Kaufmann, 1989).
- G. Dolling, M. Wegener, S. Linden, and C. Hormann, “Photorealistic images of objects in effective negative-index materials,” Opt. Express 14(5), 1842–1849 (2006). [CrossRef] [PubMed]
- A. J. Danner, “Visualizing invisibility: metamaterials-based optical devices in natural environments,” Opt. Express 18(4), 3332–3337 (2010). [CrossRef] [PubMed]
- T. Ergin, J. C. Halimeh, N. Stenger, and M. Wegener, “Optical microscopy of 3D carpet cloaks:ray-tracing calculations,” Opt. Express 18(19), 20535–20545 (2010). [CrossRef] [PubMed]
- J. L. Synge, Geometrical Mechanics and De Broglie Waves (Cambridge U. Press, 1954).
- M. Born, and E. Wolf, Principles of Optics (Pergamon, 1970).
- J. S. Desjardins, “Time-dependent geometrical optics,” J. Opt. Soc. Am. 66(10), 1042–1047 (1976). [CrossRef]
- J. Molcho and D. Censor, “A simple derivation and an example of Hamiltonian ray propagation,” Am. J. Phys. 54(4), 351–353 (1986). [CrossRef]
- P. S. J. Russell and T. A. Birks, “Hamiltonian optics of nonuniform photonic crystals,” J. Lightwave Technol. 17(11), 1982–1988 (1999). [CrossRef]
- C. Bellver-Cebreros and M. Rodriguez-Danta, “Eikonal equation from continuum mechanics and analogy between equilibrium of a string and geometrical light rays,” Am. J. Phys. 69(3), 360–367 (2001). [CrossRef]
- Y. Jiao, S. Fan, and D. A. B. Miller, “Designing for beam propagation in periodic and nonperiodic photonic nanostructures: extended Hamiltonian method,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(3), 036612 (2004). [CrossRef] [PubMed]
- D. Schurig, J. B. Pendry, and D. R. Smith, “Calculation of material properties and ray tracing in transformation media,” Opt. Express 14(21), 9794–9804 (2006). [CrossRef] [PubMed]
- D. A. Genov, S. Zhang, and X. Zhang, “Mimicking celestial mechanics in metamaterials,” Nat. Phys. 5(9), 687–692 (2009). [CrossRef]
- K. Niu, C. Song, and M.-L. Ge, “The geodesic form of light-ray trace in the inhomogeneous media,” Opt. Express 17(14), 11753–11767 (2009). [CrossRef] [PubMed]
- M. James, Pattern Recognition (John Wiley & Sons, 1988). [PubMed]
- J. L. Rodgers and W. A. Nicewander, “Thirteen ways to look at the correlation coefficient,” Am. Stat. 42(1), 59–66 (1988). [CrossRef]

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