## Semiclassical description of non magnetic cloaking

Optics Express, Vol. 16, Issue 7, pp. 4597-4604 (2008)

http://dx.doi.org/10.1364/OE.16.004597

Acrobat PDF (847 KB)

### Abstract

We present a semiclassical description of non-magnetic cloaking. The semiclassical result is confirmed by numerical simulations of a gaussian beam scattering from the cloak. Further analysis reveals that certain beams penetrate the non-magnetic cloak thereby degrading the performance.

© 2008 Optical Society of America

## 1. Introduction

2. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science **312**, 17801782 (2006). [CrossRef]

4. W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nature Photon. **1**, 224–227 (2007). [CrossRef]

2. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science **312**, 17801782 (2006). [CrossRef]

5. G. W. Milton and N. Nicorovici, “On the cloaking effects associated with anomalous localised resonance,” Proc. R. Soc. A **462**, 30273059 (2006) [CrossRef]

7. S. A. Cummer, B. Popa, D. Schurig, D. R. Smith, and J. B. Pendry, “Full-wave simulations of electromagnetic cloaking structures,” Phys. Rev. E. **74**, 036621 (2006). [CrossRef]

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

## 2. Ray dynamics of cloaking

10. U. Leonhardt, “Optical conformal mapping,” Science **312**17771780 (2006). [CrossRef]

### 2.1. Ray trajectory in vacuum

*S⃗*). In an isotropic medium, a ray is directed along the wavevector (

*k⃗*) since the Poynting vector is parallel to

*k⃗*. This is not true in the case of an anisotropic medium where

*k⃗*is not necessarily parallel to

*S⃗*. The ray trajectory in vacuum is simply a straight line, given by the equation

*θ*and distance

*r*has been defined in Fig. 1(b) (inset) and

*ρ*is the impact parameter defined with respect to the origin. This can be viewed as a statement of conservation of angular momentum for the ray and can also be obtained from Hamilton’s equations using Eq. 1 and taking

*ε*=1 and

_{r}*ε*=1.

_{θ}### 2.2. Ray trajectory in a cloaking medium

*ε*and

_{r}*ε*. For rays which avoid this region of radius

_{θ}*a*, the equation of the ray trajectory is given as

*r*≫

*a*, the rays will essentially travel in straight lines just like in vacuum. Also note that Eq. 3 implies rays with a large impact parameter essentially travel in straight lines just like vacuum. As long as the equation of ray trajectories is given by Eq. 3 the distortion in the ray trajectory as compared to vacuum is only local to the region with radius

*a*about the origin. Thus the rays in this medium have avoided the cloaked region of radius

*a*and appear like traveling in vacuum to a distant observer which is exactly the property of a cloak. The behavior of light in such a cloaking medium has been elucidated in the inset of Fig. 1(b), where the dotted red line is the trajectory in vacuum and the blue line is a trajectory which avoids a region of radius

*a*, obtained by Eq. 3 in the cloaking medium.

### 2.3. Material parameters for cloaking

*ε*and

_{r}*ε*would lead to trajectories given by Eq. 3 for rays with varying impact parameters

_{θ}*ρ*. We start by observing that the ray trajectories given by Eq.2 belong to the Hamiltonian for vacuum

*a*is the radius of the region being cloaked. We note that the cloaking medium should be invariant with respect to

*θ*because rays in any direction should avoid the circular region. The simplest choice of

*ε*and

_{r}*ε*that gives a Hamiltonian of the form as in Eq. 5 can be obtained by observing the form of the Hamiltonian for a general cylindrically anisotropic medium as in Eq. 1. In particular, the product

_{θ}*r*

^{2}

*ε*present in the denominator of one of the terms in Eq. 1, should yield (

_{r}*r*-

*a*)

^{2}as in Eq. 5, which can be achieved by

*a*and also appear to a far away observer as traveling in vacuum.

### 2.4. Cloaking device

*a*and outer radius

*b*), refraction at the boundary between the device and vacuum will have to be taken into consideration for the smooth bending of the rays around the cloaked region. Reflections will also have to be completely avoided or minimized by matching the impedance on the outer surface of the cloak. Retaining the functional form of the cloaking medium dielectric permittivities, we construct the cloaking device using

*c*

_{1}and

*c*

_{2}are constants to be determined. The cloak Hamiltonian now becomes

*ρ*=

*b*sin(

*θ*), where

_{i}*θ*is the incident angle, we obtain the equation of the ray trajectory inside the cloak as

_{i}*θ*

_{0}is a constant evaluated by conserving angular momentum at the point of impact of the ray with the cloak.

*c*

_{1}and

*c*

_{2}have to be chosen such that the equation of every ray striking the cloak would be of the form as in Eq. 3. For that we require, from Eq. 9.

## 3. Numerical simulations

*r*=

*b*) to 0 on the inner surface (

*r*=

*a*) within which the object is placed. Here we try to understand how a gaussian beam scattering from the cloak compares with the above presented semiclassical ray trajectories. As in [11

11. Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Semiclassical theory of the hyperlens,” J. Opt. Soc. Am. A **24**, A52–A59 (2007) [CrossRef]

*B*and

_{z}*ε*

^{-1}

_{θ}

*∂B*/

_{z}*∂r*where the

*z*-axis is directed along the axis of the cylinder.

### 3.1. Reflection coefficient of the cloak

*m*angular momentum mode (cylindrical coordinates) is written for the tangential component of the electric field

^{th}*E*and the normal magnetic field

_{θ}*B*

_{z}*R*′

_{1}(

*r*), in a cylindrical anisotropic medium satisfies the following equation,

*k*

_{0}=

*ω*/

*c*. Note that exterior to the cloak and in the cloaked inner region filled with vacuum, the radial part of the axial magnetic field,

*R*

_{1}(

*r*), satisfies the standard Bessel equation and hence we can write the incident, transmitted (into the cloaked region) and scattered field in terms of Hankel functions (

*H*

^{+}

_{m},

*H*

^{-}

_{m}) where + corresponds to an outgoing cylindrical wave and ‒ to an incoming cylindrical wave.

*r*≥

*b*we have,

*r*≤

*a*

*T*is the transfer matrix for the

_{m}*m*angular momentum mode, evaluated numerically. A good check for the accuracy of the evaluated transfer matrix is the fact that

^{th}*r*and

_{m}*t*which therefore need to be evaluated accurately. To avoid the singularity at the origin inherent in our hankel function basis we demand that

_{m}## 4. Results and discussion

### 4.1. Gaussian beam

*r*=1:1λ, outer radius

_{min}*r*=5:1λ, λ=365nm and use the functional form of dielectric permittivities as in Eq. 13. The radial permittivity varies from 1 on the outer interface to 0 on the inner, while the tangential permittivity is constant and different from 1. Rays striking the cloak can be represented as a gaussian beam scattering from the cloak with some impact parameter

_{max}*p*. From the simulation result in Fig. 2(b) it is evident that the beam bends around the inner hollow region of the cloak and proceeds in the same direction it is incident from. For comparison we also show a gaussian beam traveling in vacuum in Fig. 2(a). There are reflections at each interface due to the reduced parameter implementation of the cloak. Losses and reflections are the main source of degradation of performance of the non magnetic cloak [7

7. S. A. Cummer, B. Popa, D. Schurig, D. R. Smith, and J. B. Pendry, “Full-wave simulations of electromagnetic cloaking structures,” Phys. Rev. E. **74**, 036621 (2006). [CrossRef]

*h*=

*b*-

*a*≈

*b*). Shown also in the figure is the ray trajectory evaluated using the analytical expression. The excellent agreement of the ray trajectory and the path of the center of the gaussian beam validates the semiclassical description.

### 4.2. Compression

2. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science **312**, 17801782 (2006). [CrossRef]

*r*<

*a*) to the annulus (

*a*<

*r*<

*b*). This compression is thus evident in the field of a gaussian beam being bent around the inner region [Fig. 3(a)]. Ray calculations using the semiclassical approach show the same behaviour [Fig. 3(b)].

### 4.3. On-axis ray

## 5. Conclusion

## Acknowledgements

## References and links

1. | A. Greenleaf, M. Lassas, and G. Uhlmann, “On nonuniqueness for Calderons inverse problem,” Math. Res. Let. |

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

3. | 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 |

4. | W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nature Photon. |

5. | G. W. Milton and N. Nicorovici, “On the cloaking effects associated with anomalous localised resonance,” Proc. R. Soc. A |

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

7. | S. A. Cummer, B. Popa, D. Schurig, D. R. Smith, and J. B. Pendry, “Full-wave simulations of electromagnetic cloaking structures,” Phys. Rev. E. |

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

9. | W. Cai, U. K. Chettiar, A.K. Kildishev, G. W. Milton, and V. M. Shalaev, “Non-magnetic cloak without reflection,” arXiv:0707.3641v1 |

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

11. | Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Semiclassical theory of the hyperlens,” J. Opt. Soc. Am. A |

**OCIS Codes**

(160.1190) Materials : Anisotropic optical materials

(230.0230) Optical devices : Optical devices

(260.2110) Physical optics : Electromagnetic optics

**ToC Category:**

Physical Optics

**History**

Original Manuscript: January 24, 2008

Revised Manuscript: March 8, 2008

Manuscript Accepted: March 10, 2008

Published: March 19, 2008

**Citation**

Zubin Jacob and Evgenii E. Narimanov, "Semiclassical description of non magnetic cloaking," Opt. Express **16**, 4597-4604 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-7-4597

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### References

- A. Greenleaf, M. Lassas, and G. Uhlmann, "On nonuniqueness for Calderons inverse problem," Math. Res. Let. 10, 685-693 (2003).
- J. B. Pendry, D. Schurig, and D. R. Smith, "Controlling electromagnetic fields," Science 312, 17801782 (2006). [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, 977980 (2006).
- W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, "Optical cloaking with metamaterials," Nature Photon. 1, 224-227 (2007). [CrossRef]
- G. W. Milton and N. Nicorovici, "On the cloaking effects associated with anomalous localised resonance," Proc. R. Soc. A 462, 30273059 (2006). [CrossRef]
- A. Alu and N. Engheta, "Achieving transparency with plasmonic and metamaterial coatings," Phys. Rev. E. 72, 016623 (2005). [CrossRef]
- S. A. Cummer, B. Popa, D. Schurig, D. R. Smith, and J. B. Pendry, "Full-wave simulations of electromagnetic cloaking structures," Phys. Rev. E. 74, 036621 (2006). [CrossRef]
- D. Schurig, J. B. Pendry, and D. R. Smith, "Calculation of material properties and ray tracing in transformation media," Opt. Express 14, 9794 (2006). [CrossRef] [PubMed]
- W. Cai, U. K. Chettiar, A. K. Kildishev, G. W. Milton, and V. M. Shalaev, "Non-magnetic cloak without reflection," arXiv:0707.3641v1.
- U. Leonhardt, "Optical conformal mapping," Science 31217771780 (2006). [CrossRef]
- Z. Jacob, L. V. Alekseyev, and E. Narimanov, "Semiclassical theory of the hyperlens," J. Opt. Soc. Am. A 24, A52-A59 (2007). [CrossRef]

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