## Cloaking a sensor for three-dimensional Maxwell’s equations: transformation optics approach |

Optics Express, Vol. 19, Issue 21, pp. 20518-20530 (2011)

http://dx.doi.org/10.1364/OE.19.020518

Acrobat PDF (1022 KB)

### Abstract

The ideal transformation optics cloaking is accompanied by shielding: external observations do not provide any indication of the presence of a cloaked object, nor is any information about the fields outside detectable inside the cloaked region. In this paper, a transformation is proposed to cloak three-dimensional objects for electromagnetic waves in sensor mode, i.e., cloaking accompanied by degraded shielding. The proposed transformation tackles the difficulty caused by the fact that the lowest multipole in three-dimensional electromagnetic radiation is dipole rather than monopole. The loss of the surface impedance of the sensor plays an important role in determining the cloaking modes: ideal cloaking, sensor cloaking and resonance.

© 2011 OSA

## 1. Introduction

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

9. W. X. Jiang, T. J. Cui, G. X. Yu, X. Q. Lin, Q. Cheng, and J. Y. Chin, “Arbitrarily elliptical-cylindrical invisible cloaking,” J. Phys. D: Appl. Phys.41, 085504 (2008). [CrossRef]

10. A. Alù and N. Engheta, “Cloaking a sensor,” Phys. Rev. Lett. **102**, 233901 (2009). [CrossRef] [PubMed]

11. A. Alù and N. Engheta, “Cloaked near-field scanning optical microscope tip for noninvasive near-field imaging,” Phys. Rev. Lett. **105**, 263906 (2010). [CrossRef]

12. A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, “Cloaking a sensor via transformation optics,” Phys. Rev. E **83**, 016603 (2011). [CrossRef]

12. A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, “Cloaking a sensor via transformation optics,” Phys. Rev. E **83**, 016603 (2011). [CrossRef]

12. A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, “Cloaking a sensor via transformation optics,” Phys. Rev. E **83**, 016603 (2011). [CrossRef]

14. G. Castaldi, I. Gallina, V. Galdi, A. Alù, and N. Engheta, “Analytical study of spherical cloak/anti-cloak interactions,” Wave Motion **48**, 455–467 (2011). [CrossRef]

**83**, 016603 (2011). [CrossRef]

## 2. Configuration of the cloak and the sensor

*R*

_{0}, with surface impedance boundary condition −

*E*/

_{θ}*H*=

_{ϕ}*E*/

_{ϕ}*H*=

_{θ}*α*

_{0}(see [15

15. Y.-L. Geng, C.-W. Qiu, and N. Yuan, “Exact solution to electromagnetic scattering by an impedance sphere coated with a uniaxial anisotropic layer,” *IEEE Trans. Antennas Propagat.*57, 572–576 (2009). [CrossRef]

*R*

_{1}and

*R*

_{2}, respectively. All three spheres mentioned above are concentric, with the center being the origin of the coordinate system in the physical space. The space between spheres of radii

*R*

_{0}and

*R*

_{1}, as well as the space outside of the sphere of radius

*R*

_{2}, are free space. The permittivity and permeability of the cloak layer are obtained using the invariance of Maxwell’s equations under transformation of the spatial coordinate systems. Consider a coordinate transformation between the virtual space (curved free space) with the physical space. The former has spatial coordinates

*r*′,

*θ*′,

*ϕ*′, permittivity

*ɛ*

_{0}, and permeability

*μ*

_{0}, whereas the latter has spatial coordinates

*r*,

*θ*,

*ϕ*, and parameters

*ɛ*

*̿*,

*μ*

*̿*. The transformation is only in the radial direction, i.e., The associated permittivity and permeability tensors are given by where

*f*(

*R*

_{1}) = 0 and

*f*(

*R*

_{2}) =

*R*

_{2}yields perfect invisibility. To avoid singularities, we let the boundary of the inner cloaking material be at

*R*

_{1}+

*δ*, where

*δ*is a small positive number. The incident wave is generated by a time-harmonic [exp( − i

*ω*

*t*)] source that is located outside of the sphere of radius

*R*

_{2}. To achieve cloaking a sensor, we aim at obtaining a negligible scattered field outside of the sphere of radius

*R*

_{2}and at the same time perceivable electromagnetic field at the surface of the cladding sphere.

## 3. Analytical results for electromagnetic fields

**B**field can be expressed as where the scalar potential Φ

*satisfies where*

_{M}*N*is the highest order multipole used in the numerical simulations,

*B̂*(

_{n}*z*) =

*zb*(

_{n}*z*) is the Riccati-Bessel function, and

**H**and the electric field

**E**can be expressed as

*in the three regions is written as Since the source is given,*

_{M}*K*is uniquely determined. For each multipole of order

_{nm}*n*, we will solve for five unknows

*A*,

_{nm}*B*,

_{nm}*C*,

_{nm}*D*, and

_{nm}*E*. Eqs. (9) and (10) indicate that the continuities of tangential components of

_{nm}**H**and

**E**across the boundaries

*r*=

*R*

_{1}+

*δ*and

*r*=

*R*

_{2}amount to the continuities of Φ

*and*

_{M}*α*is proportional to

*α*

_{0}. Thus, we have the following boundary conditions

*B*=

_{nm}*K*and

_{nm}*A*=

_{nm}*C*. There are two cases for Eq. (18).

_{nm}### 3.1. Case 1

*Ĵ*′

*(*

_{n}*k*

_{0}

*R*

_{0}) –

*α*

*Ĵ*(

_{n}*k*

_{0}

*R*

_{0}) ≠ 0, we have

*A*and

_{nm}*E*from two linear equations,

_{nm}*E*without using any approximation. From here onwards, we will use the fact that

_{nm}*δ*is a small parameter to simplify the obtained analytical result. For an infinitesimal parameter

*z*, we have the asymptotics

*Ĵ*(

_{n}*z*) ≈

*p*

_{n}*z*

^{n}^{+1}and

*f*(

*R*

_{1}) = 0, we obtain

*O*(·) denotes terms of the same order, i.e., neglecting constant multipliers and higher-order terms. Depending on whether or not the leading term in

*F*is zero, we discuss the two following two cases.

_{n}#### 3.1.1. Case 1.1

*E*, we can express the transform function

_{nm}*f*(

*R*

_{1}+

*δ*) as, considering the fact that

*f*(

*R*

_{1}) = 0 and

*δ*is small, for some

*β*and

*s*, where

*s*> 0. Note that the Landau little-o notation

*o*(·) denotes higher order terms, i.e., those approaching to zero at faster rates. Thus, From Eq. (23), a straightforward calculation gives that

*A*= (

_{nm}*O*

*δ*

^{(2}

^{n}^{+1)}

*). Since*

^{s}*n*≥ 1, we see both

*E*and

_{nm}*A*approach zero as

_{nm}*δ*approaches zero, indicating that cloaking effect is coupled with shielding effect, i.e., there is no sensor effect.

#### 3.1.2. Case 1.2

*k*

_{0}

*R*

_{1}yields

*A*

_{2}≠ 0,

*A*

_{3}≠ 0, and

*A*

_{4}=

*A*

_{5}= 0. Due to the presence of a nonzero

*A*

_{3},

*F*contains a term of order

_{n}*O*(

*δ*

^{−}

*), which is infinite and eventually leads to zero value of*

^{ns}*E*as

_{nm}*δ*goes to zero. In this case, there is no sensing effect. To achieve a non-vanishing

*E*, we have to eliminate

_{nm}*A*

_{3}and the only way is to use

*A*

_{2}to cancel it.

*A*

_{2}+

*A*

_{3}=

*o*(

*δ*) can be satisfied when −

^{s}*n*

*δ*–

*βδ*= 0, i.e., The condition that

^{s}*n*

_{0}, they cannot be simultaneously satisfied by others orders. For

*n*=

*n*

_{0}, we easily obtain that

*E*=

_{nm}*O*(

*δ*

^{n0–2}) and

*A*=

_{nm}*O*(

*δ*

^{2n0–1}); For

*n*≠

*n*

_{0}, we obtain from Case 1.1 that

*E*=

_{nm}*O*(

*δ*

^{n}^{+1}) and

*A*=

_{nm}*O*(

*δ*

^{2}

^{n}^{+1}). The behaviors of cloaking and shielding effect are different for various order of multipoles.

- Case of
*n*_{0}= 1:*E*_{1}=_{m}*O*(*δ*^{−1}) approaches infinity and*A*_{1}=_{m}*O*(*δ*^{1}) approaches zero, which means the resonance mode. It is worth mentioning that the interior resonance does not destroy the cloaking effect, which is different from the conclusion of [12**83**, 016603 (2011). [CrossRef]**83**, 016603 (2011). [CrossRef] - Case of
*n*_{0}= 2:*E*_{2}=_{m}*O*(*δ*^{0}) is in the same order as the incidence wave and*A*_{2}=_{m}*O*(*δ*^{3}) approaches zero, which means sensor mode. - Case of
*n*_{0}≥ 3: Both*E*and_{nm}*A*approach zero, which means ideal cloaking mode, i.e., cloaking effect and shielding effect are hand in hand._{nm}

*g*(

*α*) in Eq. (19), we easily obtain the value of

*α*that leads to the resonance mode (for

*n*

_{0}= 1) and the sensor mode (for

*n*

_{0}= 2),

### 3.2. Case 2

*Ĵ*(

_{n}*k*

_{0}

*R*

_{0}) –

*α*

*Ĵ*(

_{n}*k*

_{0}

*R*

_{0}) = 0, we find that

*E*= 0 and

_{nm}*α*=

*Ĵ*′

*(*

_{n}*k*

_{0}

*R*

_{0})/

*Ĵ*(

_{n}*k*

_{0}

*R*

_{0}). We carry out an analysis similar to that of Section 3.1, simply replacing

*Ĵ*(

_{n}*k*

_{0}

*R*

_{1})

*D*. Thus, the necessary conditions for an electromagnetic wave to penetrate the inner boundary of the cloak are

_{nm}*Ĵ*(

_{n}*k*

_{0}

*R*

_{1}) = 0 and

*f*(

*R*

_{1}+

*δ*) = –

*n*

*δ*+

*o*(

*δ*) for a particular

*n*

_{0}. The resonance mode (for

*n*

_{0}= 1) and the sensor mode (for

*n*

_{0}= 2) are the same as those discussed in Section 3.1.

### 3.3. Removal of singularity

*f*(

*R*

_{1}+

*δ*) = −

*n*

*δ*+

*o*(

*δ*). We note that the condition of zero scattering outside of the cloak requires that

*f*(

*R*

_{2}) =

*R*

_{2}. Thus, if

*r*′ =

*f*(

*r*) is a continuous function, inevitably there is a particular value of

*r*, say

*R*, for which

_{b}*r*′ = 0. Consequently, from Eq. (12) we conclude that a singularity appears in the cloak layer since the argument for

*f*to be discontinuous at

*r*=

*R*, but at the same time, we keep the continuity of the permittivity and permeability. With this purpose, we see from Eqs. (4) and (5) that a function

_{b}*f*(

*r*) that simultaneously satisfies

*f*is depicted in Fig. 1.

*r*′ =

*f*(

*r*) for the case

*r*′ < 0. In deriving the permittivity and permeability in the physical space (Eqs. (4) and (5)), the key step is to use

*x*′/

*x*=

*y*′

*/y*=

*z*′/

*z*=

*r*′/

*r*[3

3. Y. Luo, H. S. Chen, J. J. Zhang, L. X. Ran, and J. A. Kong, “Design and analytical full-wave validation of the invisibility cloaks, concentrators, and field rotators created with a general class of transformations,” Phys. Rev. B **77**, 125127 (2008). [CrossRef]

17. D. Schurig, J. B. Pendry, and D. R. Smith, “Calculation of material properties and ray tracing in transformation media,” Opt. Express **14**, 9794–9804 (2006). [CrossRef] [PubMed]

*r*′ < 0, a point in the physical space and the corresponding point in the virtual space are on the opposite side of the origin. The invisibility conditions

*f*(

*R*

_{1}) = 0 and

*f*(

*R*

_{2}) =

*R*

_{2}indicate that as we move from the inner boundary of the cloak to the outer one in the physical space, the corresponding point in the virtual space moves from the original to

*r*′ =

*R*

_{2}. However, since the virtual space is free space, there is no scattering at all. Consequently, it does not matter in which way we move from the original to

*r*′ =

*R*

_{2}in the virtual space. The linear transform that is presented in [1

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

17. D. Schurig, J. B. Pendry, and D. R. Smith, “Calculation of material properties and ray tracing in transformation media,” Opt. Express **14**, 9794–9804 (2006). [CrossRef] [PubMed]

3. Y. Luo, H. S. Chen, J. J. Zhang, L. X. Ran, and J. A. Kong, “Design and analytical full-wave validation of the invisibility cloaks, concentrators, and field rotators created with a general class of transformations,” Phys. Rev. B **77**, 125127 (2008). [CrossRef]

## 4. Numerical simulations

*R*

_{0}= 0.5

*λ*,

*R*

_{1}= 1.5

*λ*,

*R*

_{2}= 3.0

*λ*,

*R*= 2.0

_{a}*λ*,

*R*= 2.33

_{b}*λ*,

*R*= 2.66

_{c}*λ*, and

*h*= 0.15

*λ*. The transform function in the range of

*R*

_{1}+

*δ*≤

*r*≤

*R*

_{2}is given by

*ɛ*

*̿*and

*μ*

*̿*are both negative and finite in the region

*R*

_{1}+

*δ*≤

*r*<

*R*, and the radial components approach zero at the boundary

_{a}*r*=

*R*

_{1}+

*δ*. In other regions, both

*ɛ*

*̿*and

*μ*

*̿*are positive and finite. It is important to stress that although four transformations are used in the region of

*R*

_{1}+

*δ*≤

*r*≤

*R*

_{2}, the potential, and consequently the electromagnetic field, is expressed in a single formula, Eq. (12). The surface impedance

*α*is calculated through Eq. (37).

*x*-polarized plane wave with a unit amplitude

**E**

^{inc}=

*x̂e*

^{ik0z}is incident upon the cloaking device along the

*z*direction. Since the given incidence wave contains only the |

*m*| = 1 term, from here onwards, we drop off the

*m*in the subscript for simplicity. For example, the coefficient

*K*is written as

_{nm}*K*. It is well known that

_{n}*K*is proportional to

_{n}*i*(2

^{n}*n*+ 1)/[

*n*(

*n*+1)] for both TE and TM components of the incident plane wave [3

3. Y. Luo, H. S. Chen, J. J. Zhang, L. X. Ran, and J. A. Kong, “Design and analytical full-wave validation of the invisibility cloaks, concentrators, and field rotators created with a general class of transformations,” Phys. Rev. B **77**, 125127 (2008). [CrossRef]

*δ*approaches zero. The cloaking effect is quantified by |

*A*/

_{n}*K*|. The lower the value of |

_{n}*A*/

_{n}*K*|, the better the cloaking effect. Since Eq. (19) shows that

_{n}*E*/

_{n}*D*is independent of

_{n}*δ*, we can use only |

*D*/

_{n}*K*| to quantify the penetrating effect. The lower the value of |

_{n}*D*/

_{n}*K*|, the poorer the penetrating ability. The case of |

_{n}*D*/

_{n}*K*| ≫ 1 corresponds to the resonance effect.

_{n}*n*

_{0}= 1. The quantities of |

*A*/

_{n}*K*| and |

_{n}*D*/

_{n}*K*| for different values of the small parameter

_{n}*δ*and order number

*n*are shown in Fig. 2. We see from Fig. 2(a) that the cloaking effect applies to all orders, including

*n*=

*n*

_{0}= 1. In addition, the slopes of the curves, which are in the logarithmic scale, agree with the theory presented in Section 3. That is to say, whenever

*δ*is decreased by a factor 10, the values of |

*A*/

_{n}*K*| decreases by a factor of 10, 10

_{n}^{5}, and 10

^{7}for

*n*= 1,

*n*= 2, and

*n*= 3, respectively. We see from Fig. 2(b) that the resonance effect applies to the order

*n*=

*n*

_{0}= 1 and the shielding effect applies to

*n*= 2 and

*n*= 3. In addition, the slopes of the curves agree with the theory, i.e., whenever

*δ*is decreased by a factor 10, the values of |

*D*/

_{n}*K*| increase by a factor of 10 for

_{n}*n*=

*n*

_{0}= 1 and decreases by a factor of 10

^{3}and 10

^{4}for

*n*= 2 and

*n*= 3, respectively. To summarize, for

*n*

_{0}= 1, the wave component corresponding to

*n*=

*n*

_{0}yields resonance effect without external scattering, whereas the wave component corresponding to

*n*≠

*n*

_{0}simultaneously yields cloaking and shielding effect.

*A*/

_{n}*K*| and |

_{n}*D*/

_{n}*K*| in the case when the real surface impedance

_{n}*α*is replaced by a complex one

*α*(1 +

*iL*), where

_{t}*L*denotes loss tangent. Fig. 3 depicts the values of |

_{t}*A*/

_{n}*K*| and |

_{n}*D*/

_{n}*K*| for

_{n}*δ*= 10

^{−3}for the range of loss tangent from 10

^{−10}to 10

^{−1}. Fig. 3(a) and (b) show that the loss enhances the cloaking effect and at the same time decreases the penetrating effect for the order

*n*=

*n*

_{0}= 1, but it barely affects the cloaking or penetrating effect for other orders. We see from Fig. 3(b) that for

*n*=

*n*

_{0}= 1, as the loss decreases, the penetrating ability changes from the shielding mode (i.e., ideal cloaking mode) to the sensor mode and then to the resonance mode. It is interesting to see that a moderate loss, such as loss tangent of 10

^{−4}, is able to enhance the cloaking effect and at the same time to shift the resonance mode to the sensor mode. In Fig. 4, we plot the

*x*component of the electric field in the

*xz*plane for the loss tangent of 10

^{−2}, 10

^{−4}, and 10

^{−7}, corresponding to the shielding (i.e., ideal cloaking), sensor, and resonance mode, respectively.In comparison,when only the sensor exists, without the presence of the outer cloaking layer, the distribution of the electric field is plotted in Fig. 5, where we see that the distorted field pattern indicates the presence of a scatterer.

*n*

_{0}= 2, we also analyze the cloaking and penetrating effects. We see from Fig. 6(a) that the cloaking effect applies to all orders, including

*n*=

*n*

_{0}= 2 and the slopes of the curves agree with the theories, i.e., whenever

*δ*is decreased by a factor 10, the values of |

*A*/

_{n}*K*| decreases by a factor of 10

_{n}^{3}, 10

^{3}, and 10

^{7}for

*n*= 1,

*n*= 2, and

*n*= 3, respectively. We see from Fig. 6(b) that the sensor mode applies to the order

*n*=

*n*

_{0}= 2 and the shielding effect applies to

*n*= 1 and

*n*= 3. The slopes of the curves agree with the theory, i.e., whenever

*δ*is decreased by a factor 10, the values of |

*D*/

_{n}*K*| keeps the same order for

_{n}*n*=

*n*

_{0}= 2 and decreases by a factor of 10

^{2}and 10

^{4}for

*n*= 1 and

*n*= 3, respectively. To summarize, for

*n*

_{0}= 2, the wave component corresponding to

*n*=

*n*

_{0}yields sensor effect without external scattering, whereas the wave component corresponding to

*n*≠

*n*

_{0}simultaneously yields cloaking and shielding effects. Fig. 7 depicts the values of |

*A*/

_{n}*K*| and |

_{n}*D*/

_{n}*K*| for

_{n}*δ*= 10

^{−3}for presence of loss in the surface impedance. We observe that the loss enhances the cloaking effect and in the meanwhile decrease the penetrating effect for the order

*n*=

*n*

_{0}= 2, but it barely affects the cloaking or penetrating effect for other orders. We see from Fig. 7(b) that for

*n*=

*n*

_{0}= 2, as the loss decreases, the penetrating ability changes from the shielding mode to the sensor mode. In Fig. 8, we plot the

*x*component of the electric field in the

*xz*plane for the loss tangent of 10

^{−4}and 10

^{−7}, corresponding to the shielding and the sensor mode, respectively. In practice, for

*n*=

*n*

_{0}= 2, the sensor mode can easily be destroyed by the loss. For example, in Fig. 8 a loss tangent of 10

^{−4}is able to shift the sensor mode to the shielding mode.

## 5. Conclusion

*n*= 1 and to shift the sensor mode to the ideal cloaking mode for

*n*= 2. Thus, in real world applications, where loss is present on the surface of sensor, it is more desirable to achieve the sensor mode using the dipole term.

## Acknowledgments

## References and links

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

2. | A. Greenleaf, M. Lassas, and G. Uhlmann, “Anisotropic conductivities that cannot be detected by EIT,” Physiol. Measure. |

3. | Y. Luo, H. S. Chen, J. J. Zhang, L. X. Ran, and J. A. Kong, “Design and analytical full-wave validation of the invisibility cloaks, concentrators, and field rotators created with a general class of transformations,” Phys. Rev. B |

4. | A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, “Full-wave invisibility of active devices at all frequencies,” Commun. Math. Phys. |

5. | B. L. Zhang, H. S. Chen, B. I. Wu, and J. A. Kong, “Extraordinary surface voltage effect in the invisibility cloak with an active device inside,” Phys. Rev. Lett. |

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

7. | H. Chen, X. Luo, H. Ma, and C. T. Chan, “The anti-cloak,” Opt. Express |

8. | B. L. Zhang, Y. Luo, X. G. Liu, and G. Barbastathis, “Macroscopic invisibility cloak for visible light,” Phys. Rev. Lett. |

9. | W. X. Jiang, T. J. Cui, G. X. Yu, X. Q. Lin, Q. Cheng, and J. Y. Chin, “Arbitrarily elliptical-cylindrical invisible cloaking,” J. Phys. D: Appl. Phys.41, 085504 (2008). [CrossRef] |

10. | A. Alù and N. Engheta, “Cloaking a sensor,” Phys. Rev. Lett. |

11. | A. Alù and N. Engheta, “Cloaked near-field scanning optical microscope tip for noninvasive near-field imaging,” Phys. Rev. Lett. |

12. | A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, “Cloaking a sensor via transformation optics,” Phys. Rev. E |

13. | G. Castaldi, I. Gallina, V. Galdi, A. Alù, and N. Engheta, “Cloak/anti-cloak interactions,” Opt. Express |

14. | G. Castaldi, I. Gallina, V. Galdi, A. Alù, and N. Engheta, “Analytical study of spherical cloak/anti-cloak interactions,” Wave Motion |

15. | Y.-L. Geng, C.-W. Qiu, and N. Yuan, “Exact solution to electromagnetic scattering by an impedance sphere coated with a uniaxial anisotropic layer,” |

16. | M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, New York, 1972). |

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

**OCIS Codes**

(160.1190) Materials : Anisotropic optical materials

(260.2110) Physical optics : Electromagnetic optics

(290.3200) Scattering : Inverse scattering

(260.2710) Physical optics : Inhomogeneous optical media

**ToC Category:**

Physical Optics

**History**

Original Manuscript: July 27, 2011

Revised Manuscript: September 7, 2011

Manuscript Accepted: September 9, 2011

Published: October 3, 2011

**Citation**

Xudong Chen and Gunther Uhlmann, "Cloaking a sensor for three-dimensional Maxwell’s equations: transformation optics approach," Opt. Express **19**, 20518-20530 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-21-20518

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### References

- J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science312, 1780–1782 (2006). [CrossRef] [PubMed]
- A. Greenleaf, M. Lassas, and G. Uhlmann, “Anisotropic conductivities that cannot be detected by EIT,” Physiol. Measure.24, 413–419 (2003). [CrossRef]
- Y. Luo, H. S. Chen, J. J. Zhang, L. X. Ran, and J. A. Kong, “Design and analytical full-wave validation of the invisibility cloaks, concentrators, and field rotators created with a general class of transformations,” Phys. Rev. B77, 125127 (2008). [CrossRef]
- A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, “Full-wave invisibility of active devices at all frequencies,” Commun. Math. Phys.275, 749–789 (2007). [CrossRef]
- B. L. Zhang, H. S. Chen, B. I. Wu, and J. A. Kong, “Extraordinary surface voltage effect in the invisibility cloak with an active device inside,” Phys. Rev. Lett.100, 063904 (2008). [CrossRef] [PubMed]
- H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Mater.9, 387–396 (2010). [CrossRef] [PubMed]
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