## On perfect cloaking

Optics Express, Vol. 14, Issue 25, pp. 12457-12466 (2006)

http://dx.doi.org/10.1364/OE.14.012457

Acrobat PDF (141 KB)

### Abstract

We show in principle how to cloak a region of space to make its contents classically invisible or transparent to waves. The method uses sensors and active sources near the surface of the region, and could operate over broad bandwidths. A general expression is given for calculating the necessary sources, and explicit, fully causal simulations are shown for scalar waves. Vulnerability to broad-band probing is discussed, and any active scheme should detectable by a quantum probe, regardless of bandwidth.

© 2006 Optical Society of America

## 1. Introduction

1. A. Alu and N. Engheta, “Achieving transparency with plasmonic and metamaterial coatings,” Phys. Rev. E **72**, 016623 (2005). [CrossRef]

5. E. Wolf and T. Habashy, “Invisible bodies and uniqueness of the inverse scattering problem,” J. Modern Opt. **40**, 785–792 (1993). [CrossRef]

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

## 2. Cloaking and local sources

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

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

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

10. E. Friot and C. Bordier, “Real-time active suppression of scattered acoustic radiation,” J. Sound Vib. **278**, 563–580 (2004). [CrossRef]

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

10. E. Friot and C. Bordier, “Real-time active suppression of scattered acoustic radiation,” J. Sound Vib. **278**, 563–580 (2004). [CrossRef]

10. E. Friot and C. Bordier, “Real-time active suppression of scattered acoustic radiation,” J. Sound Vib. **278**, 563–580 (2004). [CrossRef]

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

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

*d*, through the volume in Fig. 1(a) is necessarily shorter than any path,

_{straight}*d*, round the surface. If we exclude any wave propagation through the volume, the soonest any wave can get to the far side, and, with purely local response, the soonest any sources on the far side can respond, is through the longer path,

_{outside}*d*. Hence it is not possible to reproduce the correct form of the pulse just on the far side of the volume. No matter what response we choose for the material or locally responding sources on the far side of the volume, the “transmitted” pulse must be delayed at least in part, leading at least to some distortion. There would always be some kind of “shadow” region in which a “transmitted” pulse is at least changed. Those changes allow us to detect any attempted cloaking based on locally responding materials or sources. This argument is changed in detail but not in substance if the cloaking material or sources have finite thickness. No locally responding material or sources can perfectly cloak the volume. Equivalently, any cloaking by any locally responding material will always give rise to some scattering.

_{outside}12. Yu. I. Bobrovnitskii, “A new solution to the problem of an acoustically transparent body,” Acoust. Phys. **50**, 647–650 (2004). [CrossRef]

## 3. Excluding scalar waves from a volume

*V*based only on surface sources deduced from local values of the field at the surface

*S*. We presume that, in the region outside the volume

*V*, the wave

*ϕ*is propagating subject to the usual homogeneous scalar wave equation ∇

^{2}

*ϕ*-(1/

*c*

^{2})(∂

^{2}

*ϕ*/∂

*t*

^{2})=0 where

*c*is the wave velocity. To exclude the wave from the volume, we can now add sources to the surface of the volume (formally giving an inhomogeneous equation). For scalar waves of local amplitude

*ϕ*, for example, point sources of strength -∂

*ϕ*/∂

*n*per unit area and dipole sources oriented perpendicular to the surface and of strength

*ϕ*per unit area on

*S*will exclude the field from the volume [8

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

13. J. A. Stratton and L. J. Chu, “Diffraction Theory of Electromagnetic Waves,” Phys. Rev. **56**, 99–107 (1939). [CrossRef]

*n*is in the direction normal to the surface and outward from

*V*. Formally, the initial conditions on this problem are the (mathematically) known initial incident field. The surface sources will formally be retarded and will obey the radiation condition. Such sources function as “perfect absorbers” for scalar waves; on a large planar surface, for example, such sources would give exactly no reflection of a normally incident plane wave.

*S*to give no wave inside

*S*, then the sources −ｐ would exactly recreate that same wave starting on the inside of the surface

*S*; this is the same as saying that the sources ｐ generate a wave that exactly cancels the original wave inside the surface

*S*. Note that such sources work for stopping and creating waves; sources constructed by the above prescription make no explicit distinction between these two processes.

*h*outside

*V*, oriented perpendicular to the surface

*S*, are aligned with corresponding pairs of sources straddling

*S*, with, for simplicity, the same separations

*s*in each pair. The difference between the measured waves at these two sensors gives the wave gradient ∂

*ϕ*/∂

*n*, and their average gives the amplitude

*ϕ*. We emulate dipole sources per unit area with appropriate opposite values on the two sources in a pair, and can also choose to emulate a point source by assigning half of its value to each of the elements in the source pair. Then we get a particularly simple formula for calculating the resulting inner and outer source amplitudes

*p*and

_{in}*p*respectively based on the inner and outer measured wave amplitudes

_{out}*f*and

_{in}*f*respectively from the corresponding sensor pair, namely

_{out}*δa*is the effective element of area on

*S*“occupied” by a given source pair.

*S*does indeed exclude all waves from the volume

*V*, as can be seen in the simulation in Fig. 2(c). But it does not make the volume invisible; the transmitted wave on the far (right) side of

*V*is significantly changed [compare the original wave with no cloaking in Fig. 2(b)]. We see the kind of “shadowing” perturbation in the field that we would have expected from the argument in Fig. 1(a). Though we could not see inside

*V*, we could deduce at least that there is something on the surface

*S*.

*V*, they do actually influence the wave outside

*V*, even though these are the sources we would expect to use for a “perfect absorber”.

## 4. Method for calculating sources for true, causal cloaking

*S*) as a vector p. The sources that eliminate the waves from inside

*V*, as in the quasi-cloaking scheme above, can then formally be deduced from the local measured wave values through a linear operator Ｃ that is local in both space and time, i.e.,

*δa/s*(where

*δa*is possibly different for different source pairs). The result ｐ=Ｃｆ would contain, in order, the values of the sources at the outer and inner source points for each source pair for each time of interest.

*δ*(|

**r**-

**r**

_{s}|-

*c*(

*t*-

*t*))/4

_{s}*π*|

**r**-

**r**

_{s}| where

**r**

_{s},

*t*,

_{s}**r**, and

*t*are respectively the source position and time, and the position and time of interest for the resulting wave.

*c*is the wave velocity.

*S*a set of sources that is chosen to be always exactly equal and opposite to the sources p, i.e., sources ｐ

_{v}=-p. Since there are now no net sources on or near the surface, the wave f is everywhere exactly as it was before, i.e., it is simply the original incident wave ｆ

_{inc}, i.e., now ｆ=ｆ

_{inc}. The sources now calculated on the basis of the local wave amplitudes, ｐ=Ｃｆ(=Ｃｆ

_{inc}), would therefore be the sources that, on their own, would exactly stop the

*incident*wave, cloaking the volume, and leaving the wave otherwise unchanged; these sources, ｐ=Ｃｆ

_{inc}are the ones that would give “predetermined” cloaking [9]. If we could deduce such source values on their own (i.e., without the cancelling sources ｐ

_{v},) as a result of causal measurements, we would therefore have solved the problem of true cloaking. The question is whether we can deduce them in a causal fashion.

_{v}, and the original wave. Hence, what we can do, instead of actually having the sources ｐ

_{v}, is merely to

*calculate*what the wave would be from the sources ｐ

_{v}, and mathematically add it to the signals from the sensors when setting the sources ｐ. The wave from such (now virtual) sources would be f

_{v}=𝖦ｐ

_{v}=-𝖦ｐ=-𝖦Ｃｆ. Adding that virtual wave to the real wave at the sensors, we therefore should set our sources for true cloaking to be, with I as the identity operator,

*calculated*amount -𝖦Ｃｆ to the measured ｆ when determining the values of the sources to put on the surface. Note this “true” cloaking calculation is non-local; operating with 𝖦 on ｐ

_{v}adds the effects from all these virtual sources at all the different points on the surface at the relevant prior times (we formally are integrating or summing over

**r**

_{s}and

*t*).

_{s}## 5. Causal simulations for scalar waves

*h*=2 units further out, at the same angles, corresponding pairs of sensing points, with pair separation along the radii of

*s*=0.1 units for sense pairs and source pairs. The pictures show the resulting wave values on the equatorial plane of the sphere. (Note that the method described here is not, however, restricted to spherical surfaces.) We need finite separation

*h*between sources and sensors because we are approximating distributed sources per unit area with lumped source pairs, and the sensors consequently have to be separated from the sources by an amount comparable to or larger than the lateral separation of the sources for such an approximation to work.

*δa*~1.54 cm

^{2}“occupied” by each source pair on the sphere surface. If we consider ordinary acoustic waves in air, with a sound velocity of ~340 m/s, then the time units are (1 cm)/(340 m/s)=29.4 µs, the time taken to propagate sound across the sphere would be 1.18 ms, and the pulse amplitude FWHM would be ~0.5 ms in this simulation, corresponding to frequency bandwidths in the kHz range.

*h*, the cloaking calculations are necessarily approximate; we measure the wave at a finite distance from

*S*, rather than exactly on

*S*. Though no information is required to flow faster than the wave velocity in this simulation, we do presume instantaneous calculation in this example. Figure 3 shows the resulting pulse as a function of time at a point

*P*(Fig. 2) that is 29.5 units to the right of the center of the volume.

*V*to stimulate the sources on the right hand side. The approximate “true-cloaking” (global calculation) case shows some small delay and a small negative tail. This delay and tail result from the finite actual delay in the wave propagating up to an additional 2 units out to the sensing points and a further 2 units for that information to propagate back to the sources. If in this simulation we set up the values of the sources on the surface based on prior knowledge of what the unperturbed wave should be on the surface (“predetermined cloaking”), then simulations with the same number of sources show essentially perfect cloaking of the volume, with a transmitted pulse indistinguishable on the scales of Figs. 2 and 3 from the original wave; the minor discrepancies for the approximate “true cloaking” in Figs. 2 and 3 do not result from the finite number of sources used, but rather from the finite separation between sensors and sources.

*V*is quite effective, and there is little if any backward scattered wave. Hence, from a practical point of view, both methods could provide good cloaking against reflective probing (as in typical sonar or radar). Both methods also do exclude the wave from the volume.

## 6. Discussion

### 6.1. Sources for cloaking of electromagnetic waves

13. J. A. Stratton and L. J. Chu, “Diffraction Theory of Electromagnetic Waves,” Phys. Rev. **56**, 99–107 (1939). [CrossRef]

**E**and

**H**components thus determined, Maxwell’s equations then determine all the changing components of the fields perpendicular to the surface, so no other sources are then required for time-varying fields. (Static electric fields can also be handled with the addition of electric charge on the surface, though static magnetic fields are more difficult because of the physical absence of magnetic monopole sources.) Hence, at least in principle, we can construct a similar operator Ｃ with real physical sources on the surface for time-varying electromagnetic fields, and the process of cloaking could otherwise proceed analogously to that above for scalar waves. Note that we now have twice as many sensors and sources (two electric and two magnetic sensors and sources at each point on the surface) as in the scalar case; we would expect this since now we have to be able to handle two distinct polarizations of fields. If we were to set up a simulation analogous to the scalar simulation above, but for electromagnetic waves in free space, and take a similar 1 cm distance unit, and hence a 40 cm diameter volume, the time unit would be ~33 ps, and the pulse width would be ~0.55 ns, corresponding to GHz bandwidths.

### 6.2. Propagation and calculation time

*V*at least as fast as the wave velocity outside. For slowly propagating waves, such as acoustic waves, we could send the information at higher speed inside

*V*along wires or optical fibers around the inside of the surface

*S*, leaving a clear, usable cloaked volume, and still have time for calculations. For electromagnetic waves in a vacuum, however, it could be difficult to make the calculation delay small compared to the time taken for light to propagate across the volume; also, to avoid excessively perforating the volume with wires, beams, or waveguides, we would certainly have additional propagation delay for information flowing through the volume. Such cloaking would be detectable in transmission with pulses of length comparable to the sum of the additional delays involved. Whether such an approach is then usable depends on the bandwidth or length of the probing signals.

### 6.3 Quantum detection of cloaking

18. N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. **74**, 145–195 (2002). [CrossRef]

19. W. K. Wootters and W. H. Zurek, “A single quantum cannot be cloned,” Nature **299**, 802 (1982). [CrossRef]

20. D. Dieks, “Communication by EPR devices,” Phys. Lett. A **92**, 271–272 (1982). [CrossRef]

20. D. Dieks, “Communication by EPR devices,” Phys. Lett. A **92**, 271–272 (1982). [CrossRef]

## 7. Conclusions

## Acknowledgments

## References and links

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

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

3. | J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling Electromagnetic Fields,” Science |

4. | G. W. Milton and N.-A. P. Nicorovici, “On the cloaking effects associated with anomalous localized resonance,” Proc. Roy. Soc. A |

5. | E. Wolf and T. Habashy, “Invisible bodies and uniqueness of the inverse scattering problem,” J. Modern Opt. |

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

7. | A. Alu and N. Engheta, “Pairing an epsilon-negative slab with a mu-negative slab: resonance, tunneling and transparency,” IEEE Trans. Antennas Propag. |

8. | J. E. Ffowcs Williams “Review Lecture: Anti-Sound,” Proc. Roy. Soc. London A |

9. | P. A. Nelson and S. J. Elliott, |

10. | E. Friot and C. Bordier, “Real-time active suppression of scattered acoustic radiation,” J. Sound Vib. |

11. | E. Friot, R. Guillermin, and M. Winninger, “Active control of scattered acoustic radiation: a real-time implementation for a three-dimensional object,” Acta Acust. |

12. | Yu. I. Bobrovnitskii, “A new solution to the problem of an acoustically transparent body,” Acoust. Phys. |

13. | J. A. Stratton and L. J. Chu, “Diffraction Theory of Electromagnetic Waves,” Phys. Rev. |

14. | J. A. Stratton, |

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

16. | D. A. B. Miller, “Huygens’s wave propagation principle corrected,” Opt. Lett. |

17. | C. H. Bennett and G. Brassard, |

18. | N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. |

19. | W. K. Wootters and W. H. Zurek, “A single quantum cannot be cloned,” Nature |

20. | D. Dieks, “Communication by EPR devices,” Phys. Lett. A |

**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: September 29, 2006

Revised Manuscript: November 21, 2006

Manuscript Accepted: November 28, 2006

Published: December 11, 2006

**Citation**

David A. B. Miller, "On perfect cloaking," Opt. Express **14**, 12457-12466 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-25-12457

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### References

- A. Alu and N. Engheta, "Achieving transparency with plasmonic and metamaterial coatings," Phys. Rev. E 72, 016623 (2005). [CrossRef]
- 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]
- G. W. Milton, and N.-A. P. Nicorovici, "On the cloaking effects associated with anomalous localized resonance," Proc. Roy. Soc. A 462, 3027-3059 (2006). [CrossRef]
- E. Wolf and T. Habashy, "Invisible bodies and uniqueness of the inverse scattering problem," J. Mod. Opt. 40, 785-792 (1993). [CrossRef]
- N. A. Nicorovici, R. C. McPhedran, and G. W. Milton, "Optical and dielectric properties of partially resonant composites," Phys. Rev. B 490, 8479-8482 (1994). [CrossRef]
- A. Alu and N. Engheta, "Pairing an epsilon-negative slab with a mu-negative slab: resonance, tunneling and transparency," IEEE Trans. Antennas Propag. 51, 2558-2571 (2003). [CrossRef]
- J. E. Ffowcs Williams "Review Lecture: Anti-Sound," Proc. Roy. Soc. London A 395, 63-88 (1984). [CrossRef]
- P. A. Nelson and S. J. Elliott, Active Control of Sound (Academic Press, London, 1992), pp. 290-293.
- E. Friot and C. Bordier, "Real-time active suppression of scattered acoustic radiation," J. Sound Vib. 278, 563-580 (2004). [CrossRef]
- E. Friot, R. Guillermin, and M. Winninger, "Active control of scattered acoustic radiation: a real-time implementation for a three-dimensional object," Acta Acust. 92, 278-288 (2006).
- Yu. I. Bobrovnitskii, "A new solution to the problem of an acoustically transparent body," Acoust. Phys. 50, 647-650 (2004). [CrossRef]
- J. A. Stratton and L. J. Chu, "Diffraction Theory of Electromagnetic Waves," Phys. Rev. 56, 99-107 (1939). [CrossRef]
- J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).
- M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).
- D. A. B. Miller, "Huygens's wave propagation principle corrected," Opt. Lett. 16, 1370-1372 (1991). [CrossRef] [PubMed]
- C. H. Bennett and G. Brassard, Proceedings of the IEEE International Conference on Computers, Systems and Signal Processing, Bangalore, India, 1984 (IEEE, New York, 1984), 175-179. [PubMed]
- N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, "Quantum cryptography," Rev. Mod. Phys. 74, 145-195 (2002). [CrossRef]
- W. K. Wootters and W. H. Zurek, "A single quantum cannot be cloned," Nature 299, 802 (1982). [CrossRef]
- D. Dieks, "Communication by EPR devices," Phys. Lett. A 92, 271-272 (1982). [CrossRef]

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