## Broadband exterior cloaking

Optics Express, Vol. 17, Issue 17, pp. 14800-14805 (2009)

http://dx.doi.org/10.1364/OE.17.014800

Acrobat PDF (1013 KB)

### Abstract

It is shown how a recently proposed method of cloaking is effective over a broad range of frequencies. The method is based on three or more active devices. The devices, while not radiating significantly, create a “quiet zone” between the devices where the wave amplitude is small. Objects placed within this region are virtually invisible. The cloaking is demonstrated by simulations with a broadband incident pulse.

© 2009 Optical Society of America

## 1. Introduction

6. M. Kerker, “Invisible bodies,” J. Opt. Soc. Am. **65**, 376–379 (1975).
[CrossRef]

7. A. Alú and N. Engheta, “Achieving transparency with plasmonic and metamaterial coatings,” Phys. Rev. E **72**, 016,623 (2005).
[CrossRef]

8. A. Greenleaf, M. Lassas, and G. Uhlmann, “Anisotropic conductivities that cannot be detected by EIT,” Physiol. Meas. **24**, 413–419 (2003).
[CrossRef] [PubMed]

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

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

14. A. N. Norris, “Acoustic cloaking theory,” Proc. R. Soc. Lon. Ser. A. Math. Phys. Sci. **464**, 2411–2434 (2008).
[CrossRef]

15. G.W. Milton, M. Briane, and J. R. Willis, “On cloaking for elasticity and physical equations with a transformation invariant form,” New J. Phys. **8**, 248 (2006).
[CrossRef]

17. M. Farhat, S. Guenneau, S. Enoch, and A. B. Movchan, “Cloaking bending waves propagating in thin elastic plates,” Phys. Rev. B **79**, 033102 (2009).
[CrossRef]

18. M. Farhat, S. Enoch, S. Guenneau, and A. B. Movchan, “Broadband cylindrical acoustic cloak for linear surface waves in a fluid,” Phys. Rev. Lett. **101**, 134,501 (2008).
[CrossRef]

18. M. Farhat, S. Enoch, S. Guenneau, and A. B. Movchan, “Broadband cylindrical acoustic cloak for linear surface waves in a fluid,” Phys. Rev. Lett. **101**, 134,501 (2008).
[CrossRef]

23. G. W. Milton and N.-A. P. Nicorovici, “On the cloaking effects associated with anomalous localized resonance,” Proc. R. Soc. Lon. Ser. A. Math. Phys. Sci. **462**, 3027–3059 (2006).
[CrossRef]

25. G. W. Milton, N.-A. P. Nicorovici, R. C. McPhedran, K. Cherednichenko, and Z. Jacob, “Solutions in folded geometries, and associated cloaking due to anomalous resonance,” New J. Phys. **10**, 115,021 (2008).
[CrossRef]

26. V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of *ε* and *µ*,” Sov. Phys. Usp. **10**, 509–514 (1968).
[CrossRef]

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

29. O. P. Bruno and S. Lintner, “Superlens-cloaking of small dielectric bodies in the quasistatic regime,” J. Appl. Phys. **102**, 124,502 (2007).
[CrossRef]

30. 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**, 093,901 (2009).
[CrossRef]

20. R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science **323**, 366–369 (2009).
[CrossRef] [PubMed]

31. J. Li and J. B. Pendry, “Hiding under the carpet: a new strategy for cloaking,” Phys. Rev. Lett. **101**, 203,901 (2008).
[CrossRef]

32. U. Leonhardt and T. Tyc, “Broadband invisibility by non-Euclidean cloaking,” Science **323**, 110–112 (2009).
[CrossRef]

2. D. A. B. Miller, “On perfect cloaking,” Opt. Express **14**, 12,457–12,466 (2006).
[CrossRef]

33. J. E. Ffowcs Williams, “Review Lecture: Anti-Sound,” Proc. R. Soc. A **395**, 63–88 (1984).
[CrossRef]

34. A. W. Peterson and S. V. Tsynkov, “Active control of sound for composite regions,” SIAM J. Appl. Math. **67**, 1582–1609 (2007).
[CrossRef]

35. I. I. Smolyaninov, V. N. Smolyaninova, A. V. Kildishev, and V. M. Shalaev, “Anisotropic metamaterials emulated by tapered waveguides: Application to optical cloaking,” Phys. Rev. Lett. **102**, 213,901 (2009).
[CrossRef]

## 2. Cloaking a single frequency

*u*+

*k*

^{2}

*u*=0. Here

*u*(

**x**,

*ω*) is the wave field,

*k*=2

*π*/

*λ*is the wavenumber and

*λ*=2

*πc*

_{0}/

*ω*is the wavelength at frequency

*ω*and at a constant propagation speed

*c*

_{0}. We would like to cloak a region in the plane from a known probing (incident) wave

*u*(

_{i}**x**,

*ω*) supported in the frequency band

*ω*

_{0}+[-

*B*/2,

*B*/2], where the central frequency is

*ω*

_{0}and the bandwidth is

*B*.

**x**|≤

*α*and assume we measure the radiation emitted by the devices on the circle |

**x**|=

*γ*>

*α*. Thus the device’s field

*u*(

_{d}**x**,

*ω*) must be so that (a’)

*u*≈-

_{d}*u*for |

_{i}**x**|≤

*α*and (b’)

*u*≈0 for |

_{d}**x**|=

*γ*.

*D*points

**x**

_{1}, …,

**x**

_{D}with |

**x**

_{j}|=

*δ*and

*α*<

*δ*<

*γ*so that the devices surround the cloaked region. Because the device’s field must solve Helmholtz equation and become small far away, we take it as a linear combination of outgoing waves emanating from the source points

**x**

_{1}, …,

**x**

_{D}with the form [1]:

*H*

^{(1)}

_{n}is the

*n*-th Hankel function of the first kind and

*θ*≡arg(

_{m}**x**-

**x**

*m*) is the angle between

**x**-

**x**

_{m}and (1,0). We seek coefficients

*b*so that (a’) holds on points of the circle |

_{m,n}**x**|=

*α*and (b’) holds on points of the circle |

**x**|=

*γ*. The control points are uniformly distributed and at most

*λ*/2 apart on each circle. The resulting linear equations are solved in the least squares sense with the Truncated Singular Value Decomposition in two steps. First we find coefficients

*b*so that (a’) holds, and second we find a correction to enforce (b’) while still satisfying (a’). (see Appendix A for more details).

_{m,n}## 3. Simulations

*u*(

**x**,

*ω*) is the transverse component of the electric field. For the numerical experiments we took a central frequency and bandwidth of 2.4GHz, a propagation speed of

*c*

_{0}=3×10

^{8}m/s and a central wavelength of

*λ*

_{0}=12.5cm. Simulations suggest a minimum of three devices are needed to cloak independently of the direction of the incoming waves. In Fig. 1 we show cloaking at the central frequency

*ω*

_{0}/(2

*π*)=2.4GHz of a region of radius

*α*=2

*λ*

_{0}(solid white circle). Here the devices are located

*δ*=10

*λ*

_{0}from the origin and invisibility is enforced at a distance

*γ*=20

*λ*

_{0}from the origin (dashed white circle). The incident wave is a point source originating at

**x**

*=(-20,0)*

_{s}*λ*

_{0}and modulated in frequency by a Gaussian truncated to the bandwidth

*ω*-

*ω*

_{0}|≤

*B*/2, and 0 otherwise. We took

*σ*=4/

*ω*

_{0}. The scatterer is a perfectly conducting “kite” obstacle [36] with homogeneous Dirichlet boundary conditions, fitting inside the cloaked region. The scattered field is computed using the boundary integral equation method in [36]. This model is also valid in other contexts, e.g. in elastodynamics for anti-plane shear waves within an isotropic elastic medium, assuming the kite is rigid and clamped.

^{-4}% of the field scattered without the devices, as measured on the dashed white circle with the

*L*

^{2}norm. We carried the same procedure for

*N*=101 frequencies in the bandwidth with similar results, as can be seen in Fig. 2. Since the bandwidth is 100% of the central frequency, broadband cloaking is possible with our approach.

_{freq}*T*], with

*T*≈132ns or the time it takes for the wave to travel 50.5

*λ*

_{0}. The devices make the incoming wave disappear when it reaches the cloaked region, and then rebuild the wave as it exits the cloaked region. This makes the object virtually undetectable.

*λ*

_{0}=25cm radius). However we have successfully cloaked regions up to 10

*λ*

_{0}=1.25m in radius as shown in Fig. 4, where we take three devices with |

**x**

_{m}|=

*δ*=5

*α*and invisibility is enforced on |

**x**|=

*γ*=10

*α*.

*𝒪*(|

**x**-

**x**

_{m}|

^{-N}) singularities near the devices. Fortunately the point devices can be replaced (with Green’s identities) by curves where the fields have reasonable amplitudes and where we can control a single-and double-layer potential [36]. These curves could be the circles suggested by the contours |

*u*|=100max

_{d}_{|x|=α}|

*u*(

_{i}**x**,

*ω*)| (in black in Fig. 1). The radius of such devices for other cloaked region radii is estimated in Fig. 5. Since the devices do not completely surround the region to be cloaked, exterior cloaking is possible at least for

*α*≤10

*λ*

_{0}.

## A. Finding the driving coefficients for the devices

**b**be a vector with the (2

*N*+1)

*D*possibly complex coefficients

*b*. Denote by

_{m,n}**p**

*(resp.*

^{α}_{j}**p**

*) the*

^{γ}_{j}*N*(resp.

^{α}*N*) control points on the circle |

^{γ}**x**|=

*α*(resp. |

**x**|=

*γ*). The coefficients

**b**, the number

*N*of terms in the expression for

*u*and the control points all depend on the frequency

_{d}*ω*. Using the form for

*u*in the text, construct a matrix

_{d}**A**of size

*N*×(2

^{α}*N*+1)

*D*and a matrix

**B**of size

*N*×(2

^{γ}*N*+1)

*D*such that

*u*(

_{d}**p**

*,*

^{α}_{j}*ω*)=(

**Ab**)

_{j}and

*u*(

_{d}**p**

*,*

^{γ}_{j}*ω*)=(

**Bb**)

_{j}. We estimate the driving coefficients as follows.

**Enforce (a’)**: Use the Truncated Singular Value Decomposition (TSVD) to find coefficients

**b**

_{0}such that

**Ab**

_{0}≈-[

*u*(

_{i}**p**

*,*

^{α}_{j}*ω*)], in the least squares sense.

**Enforce (b’) while still satisfying (a’):**Find

**z**as the solution to the least squares problem

**B**(

**b**

_{0}+

**Zz**)≈

**0**. Again the TSVD can be used for this step. Here

**Z**is a matrix with columns spanning the nullspace of

**A**(a byproduct of the SVD in the previous step).

**b**=

**b**

_{0}+

**Zz**. The heuristic for

*N*in the numerical experiments is

*N*=⌈

*k*(

*δ*-

*α*/2)⌉, where ⌈

*x*⌉ is the smallest integer larger than

*x*. The choice of cut-off singular values can be used to control how well one wants to satisfy (a’) and (b’). We used a fixed 10

^{-5}tolerance relative to the maximum amplitude of

*u*on the control points.

_{i}## Acknowledgments

## References and links

1. | F. Guevara Vasquez, G. W. Milton, and D. Onofrei, “Active exterior cloaking for the 2D Laplace and Helmholtz equations,” (2009). Accepted for publication in Phys. Rev. Lett., arXiv:0906.1544v1 [math-ph]. |

2. | D. A. B. Miller, “On perfect cloaking,” Opt. Express |

3. | R. Weder, “A rigorous analysis of high-order electromagnetic invisibility cloaks,” J. Phys. A |

4. | A. G. Ramm, “Invisible obstacles,” Ann. Polon. Math. |

5. | L. S. Dolin, “To the possibility of comparison of three-dimensional electromagnetic systems with nonuniform anisotropic filling,” Izv. Vyssh. Uchebn. Zaved. Radiofizika |

6. | M. Kerker, “Invisible bodies,” J. Opt. Soc. Am. |

7. | A. Alú and N. Engheta, “Achieving transparency with plasmonic and metamaterial coatings,” Phys. Rev. E |

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

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

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

11. | H. Chen and C. T. Chan, “Acoustic cloaking in three dimensions using acoustic metamaterials,” Appl. Phys. Lett. |

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

13. | S. A. Cummer, B.-I. Popa, D. Schurig, D. R. Smith, J. Pendry, M. Rahm, and A. Starr, “Scattering Theory Derivation of a 3D Acoustic Cloaking Shell,” Phys. Rev. Lett. |

14. | A. N. Norris, “Acoustic cloaking theory,” Proc. R. Soc. Lon. Ser. A. Math. Phys. Sci. |

15. | G.W. Milton, M. Briane, and J. R. Willis, “On cloaking for elasticity and physical equations with a transformation invariant form,” New J. Phys. |

16. | M. Brun, S. Guenneau, and A. B. Movchan, “Achieving control of in-plane elastic waves,” Appl. Phys. Lett. |

17. | M. Farhat, S. Guenneau, S. Enoch, and A. B. Movchan, “Cloaking bending waves propagating in thin elastic plates,” Phys. Rev. B |

18. | M. Farhat, S. Enoch, S. Guenneau, and A. B. Movchan, “Broadband cylindrical acoustic cloak for linear surface waves in a fluid,” Phys. Rev. Lett. |

19. | 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 |

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

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

22. | L. H. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, “Cloaking at Optical Frequencies,” (2009). ArXiv:0904.3508v1 [physics.optics]. |

23. | G. W. Milton and N.-A. P. Nicorovici, “On the cloaking effects associated with anomalous localized resonance,” Proc. R. Soc. Lon. Ser. A. Math. Phys. Sci. |

24. | N.-A. P. Nicorovici, G.W. Milton, R. C. McPhedran, and L. C. Botten, “Quasistatic cloaking of two-dimensional polarizable discrete systems by anomalous resonance,” Opt. Express |

25. | G. W. Milton, N.-A. P. Nicorovici, R. C. McPhedran, K. Cherednichenko, and Z. Jacob, “Solutions in folded geometries, and associated cloaking due to anomalous resonance,” New J. Phys. |

26. | V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of |

27. | N. A. Nicorovici, R. C. McPhedran, and G. W. Milton, “Optical and dielectric properties of partially resonant composites,” Phys. Rev. B |

28. | J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. |

29. | O. P. Bruno and S. Lintner, “Superlens-cloaking of small dielectric bodies in the quasistatic regime,” J. Appl. Phys. |

30. | 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. |

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

32. | U. Leonhardt and T. Tyc, “Broadband invisibility by non-Euclidean cloaking,” Science |

33. | J. E. Ffowcs Williams, “Review Lecture: Anti-Sound,” Proc. R. Soc. A |

34. | A. W. Peterson and S. V. Tsynkov, “Active control of sound for composite regions,” SIAM J. Appl. Math. |

35. | I. I. Smolyaninov, V. N. Smolyaninova, A. V. Kildishev, and V. M. Shalaev, “Anisotropic metamaterials emulated by tapered waveguides: Application to optical cloaking,” Phys. Rev. Lett. |

36. | D. Colton and R. Kress, |

**OCIS Codes**

(160.4760) Materials : Optical properties

(260.0260) Physical optics : Physical optics

(350.7420) Other areas of optics : Waves

**ToC Category:**

Physical Optics

**History**

Original Manuscript: July 1, 2009

Revised Manuscript: July 24, 2009

Manuscript Accepted: July 27, 2009

Published: August 5, 2009

**Citation**

Fernando Guevara Vasquez, Graeme W. Milton, and Daniel Onofrei, "Broadband exterior cloaking," Opt. Express **17**, 14800-14805 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-17-14800

Sort: Year | Journal | Reset

### References

- F. Guevara Vasquez, G. W. Milton, and D. Onofrei, "Active exterior cloaking for the 2D Laplace and Helmholtz equations," (2009). Accepted for publication in Phys. Rev. Lett., arXiv:0906.1544v1 [math-ph].
- D. A. B. Miller, "On perfect cloaking," Opt. Express 14, 1457-1466 (2006). [CrossRef]
- R. Weder, "A rigorous analysis of high-order electromagnetic invisibility cloaks," J. Phys. A 41, 065,207 (2008). [CrossRef]
- A. G. Ramm, "Invisible obstacles," Ann. Polon. Math. 90, 145-148 (2007). [CrossRef]
- L. S. Dolin, "To the possibility of comparison of three-dimensional electromagnetic systems with nonuniform anisotropic filling," Izv. Vyssh. Uchebn. Zaved. Radiofizika 4, 964-967 (1961).
- M. Kerker, "Invisible bodies," J. Opt. Soc. Am. 65, 376-379 (1975). [CrossRef]
- A. Alú and N. Engheta, "Achieving transparency with plasmonic and metamaterial coatings," Phys. Rev. E 72, 016,623 (2005). [CrossRef]
- A. Greenleaf, M. Lassas, and G. Uhlmann, "Anisotropic conductivities that cannot be detected by EIT," Physiol. Meas. 24, 413-419 (2003). [CrossRef] [PubMed]
- U. Leonhardt, "Optical conformal mapping," Science 312, 1777-1780 (2006). [CrossRef] [PubMed]
- J. B. Pendry, D. Schurig, and D. R. Smith, "Controlling electromagnetic fields," Science 312, 1780-1782 (2006). [CrossRef] [PubMed]
- H. Chen and C. T. Chan, "Acoustic cloaking in three dimensions using acoustic metamaterials," Appl. Phys. Lett. 91, 183,518 (2007).
- 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]
- S. A. Cummer, B.-I. Popa, D. Schurig, D. R. Smith, J. Pendry, M. Rahm, and A. Starr, "Scattering Theory Derivation of a 3D Acoustic Cloaking Shell," Phys. Rev. Lett. 100, 024,301 (2008). [CrossRef]
- A. N. Norris, "Acoustic cloaking theory," Proc. R. Soc. Lon. Ser. A. Math. Phys. Sci. 464, 2411-2434 (2008). [CrossRef]
- G.W. Milton, M. Briane, and J. R. Willis, "On cloaking for elasticity and physical equations with a transformation invariant form," New J. Phys. 8, 248 (2006). [CrossRef]
- M. Brun, S. Guenneau, and A. B. Movchan, "Achieving control of in-plane elastic waves," Appl. Phys. Lett. 94, 061903 (2009). [CrossRef]
- M. Farhat, S. Guenneau, S. Enoch, and A. B. Movchan, "Cloaking bending waves propagating in thin elastic plates," Phys. Rev. B 79, 033102 (2009). [CrossRef]
- M. Farhat, S. Enoch, S. Guenneau, and A. B. Movchan, "Broadband cylindrical acoustic cloak for linear surface waves in a fluid," Phys. Rev. Lett. 101, 134,501 (2008). [CrossRef]
- 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, 977-980 (2006). [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, 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, 568-571 (2009). [CrossRef] [PubMed]
- L. H. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, "Cloaking at Optical Frequencies," (2009). ArXiv:0904.3508v1 [physics.optics].
- G. W. Milton and N.-A. P. Nicorovici, "On the cloaking effects associated with anomalous localized resonance," Proc. R. Soc. Lon. Ser. A. Math. Phys. Sci. 462, 3027-3059 (2006). [CrossRef]
- N.-A. P. Nicorovici, G.W. Milton, R. C. McPhedran, and L. C. Botten, "Quasistatic cloaking of two-dimensional polarizable discrete systems by anomalous resonance," Opt. Express 15, 6314-6323 (2007). [CrossRef] [PubMed]
- G. W. Milton, N.-A. P. Nicorovici, R. C. McPhedran, K. Cherednichenko, and Z. Jacob, "Solutions in folded geometries, and associated cloaking due to anomalous resonance," New J. Phys. 10, 115,021 (2008). [CrossRef]
- V. G. Veselago, "The electrodynamics of substances with simultaneously negative values of ? and ?," Sov. Phys. Usp. 10, 509-514 (1968). [CrossRef]
- N. A. Nicorovici, R. C. McPhedran, and G. W. Milton, "Optical and dielectric properties of partially resonant composites," Phys. Rev. B 49, 8479-8482 (1994). [CrossRef]
- J. B. Pendry, "Negative refraction makes a perfect lens," Phys. Rev. Lett. 85, 3966-3969 (2000). [CrossRef] [PubMed]
- O. P. Bruno and S. Lintner, "Superlens-cloaking of small dielectric bodies in the quasistatic regime," J. Appl. Phys. 102, 124,502 (2007). [CrossRef]
- 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, 093,901 (2009). [CrossRef]
- J. Li and J. B. Pendry, "Hiding under the carpet: a new strategy for cloaking," Phys. Rev. Lett. 101, 203,901 (2008). [CrossRef]
- U. Leonhardt and T. Tyc, "Broadband invisibility by non-Euclidean cloaking," Science 323, 110-112 (2009). [CrossRef]
- J. E. F. Williams, "Review Lecture: Anti-Sound," Proc. R. Soc. A 395, 63-88 (1984). [CrossRef]
- A. W. Peterson and S. V. Tsynkov, "Active control of sound for composite regions," SIAM J. Appl. Math. 67, 1582-1609 (2007). [CrossRef]
- I. I. Smolyaninov, V. N. Smolyaninova, A. V. Kildishev, and V. M. Shalaev, "Anisotropic metamaterials emulated by tapered waveguides: Application to optical cloaking," Phys. Rev. Lett. 102, 213,901 (2009). [CrossRef]
- D. Colton and R. Kress, Inverse acoustic and electromagnetic scattering theory, vol. 93 of Applied Mathematical Sciences, 2nd ed. (Springer-Verlag, Berlin, 1998).

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