## Monostatic lidar/radar invisibility using coated spheres

Optics Express, Vol. 16, Issue 3, pp. 1431-1439 (2008)

http://dx.doi.org/10.1364/OE.16.001431

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### Abstract

The Lorenz-Mie theory is revisited to explicitly include materials whose permeability is different from unity. The expansion coefficients of the scattered field are given for light scattering by both homogeneous and coated spheres. It is shown that the backscatter is exactly zero if the impedance of the spherical particles is equal to the intrinsic impedance of the surrounding medium. If spherical particles are sufficiently large, the zero backscatter can be explained as impedance matching using the asymptotic expression for the radar backscattering cross section. In the case of a coated sphere, the shell can be regarded as a cloak if the product of the thickness and the imaginary part of the refractive index of the outer shell is large.

© 2008 Optical Society of America

## 1. Introduction

8. H. Chen, B. I. Wu, B. Zhang, and J. A. Kong, “Electromagnetic Wave Interactions with a Metamaterial Cloak,” Phys. Rev. Lett. **99**, 063903-1–4 (2007). [CrossRef] [PubMed]

12. G. Mie, “Beigrade zur optik truber medien, speziell kolloidaler metallosungen,” Ann. Phys. (Leipzig) **25**, 377–455 (1908). [CrossRef]

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

*ε*/

_{t}*ε*

_{0}=

*R*

_{2}/(

*R*

_{2}-

*R*

_{1}),

*ε*=

_{r}*ε*(

_{t}*r*-

*R*

_{1})

^{2}/

*r*

^{2},

*μ*/

_{t}*μ*

_{0}=

*ε*/

_{t}*ε*

_{0}, and

*μ*/

_{r}*μ*=

_{t}*ε*/

_{r}*ε*;

_{t}*R*

_{1}and

*R*

_{2}are the inner and outer radii of the two concentric spheres, respectively;

*ε*

_{0}and

*μ*

_{0}are the permittivity and permeability of the surrounding medium;

*r̂*,

*, and*θ ^

*are the unit vectors of the spherical coordinates. It is worthy noting that the medium defined by Eq. (1) is lossless. If loss exists, the cloak becomes visible and only the exact backscatter is zero. Chen et al. [8*ϕ ^

8. H. Chen, B. I. Wu, B. Zhang, and J. A. Kong, “Electromagnetic Wave Interactions with a Metamaterial Cloak,” Phys. Rev. Lett. **99**, 063903-1–4 (2007). [CrossRef] [PubMed]

*η*

_{1}=

*η*

_{0}, which is also called the impedance matching condition, for a homogeneous spherical particle to achieve zero backscatter, where

*η*=

_{i}*η*

_{0}is satisfied for each

*i*layer of the coated sphere. Furthermore, one can speculate from physical intuition that the backscatter of a coated sphere is still approximately zero even if the impedance matching condition is violated for the inner core, given that its outer layer satisfies the condition and the imaginary part of its refractive index is sufficiently large. This is because the electromagnetic wave cannot penetrate into the inner region. Thus no matter what is inside, as long as it is a passive medium, the scattering pattern is essentially the same. This speculation is proved by numerical study and it can be used to make a cloak invisible for monostatic lidar/radar detection. The condition of this type of cloak is not as strict as that suggested in Ref. [1

^{th}1. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling Electromagnetic Fields,” Science **312**, 1780–1782 (2006). [CrossRef] [PubMed]

13. J. Maxwell Garnett, “Colours in metal glasses and in metallic films,” Philos. Trans. R. Soc. London **203**, 385 (1904). [CrossRef]

14. J.R. Liu, M. Itoh, and K.-I Machida, “Frequency dispersion of complex permeability and permittivity on ironbased nanocomposites derived from rare earth-iron intermetallic compounds,” J. Alloys Compd. **408–412**, 1396–1399 (2006). [CrossRef]

## 2. Theory and formulas

*x*is the size parameter of the sphere;

*k*is the magnitude of the wave vector in the surrounding medium;

*a*is the radius of the sphere; m is the relative refractive index of the sphere;

*ε*is the permittivity;

*μ*is the permeability. Consider the scattering of a x-polarized electromagnetic plane wave by this sphere. The propagation direction of the incident wave is along the

*z*axis. The incident, scattered, and internal fields can be expanded in terms of the vector spherical harmonics. The expansion coefficients are determined by the boundary conditions. Eqs. (4.53) in Ref. [16] gave the general expressions of the expansion coefficients of the scattered wave. Bohren and Huffman [16] assumed

*μ*

_{1}/

*μ*

_{0}=1 and obtained Eqs. (4.88) in Ref. [16] for numerical calculations. To derive the corresponding equations of Eqs. (4.88) in Ref. [16] for the case of

*μ*

_{1}/

*μ*

_{0}≠1, the relative impedance

*η*is introduced:

*ψ*(

_{n}*ρ*)=

*ρj*(

_{n}*ρ*) and

*ξ*(

_{n}*ρ*)=

*ρh*

^{(1)}

*(*

_{n}*ρ*) are the Riccati-Bessel functions;

*j*and

_{n}*h*

^{(1)}

*=*

_{n}*j*+

_{n}*iy*are the spherical Bessel functions of the first and third kinds, respectively;

_{n}*y*is the spherical Bessel function of the second kind.

_{n}*D*(

_{n}*ρ*)=

*d*ln

*ψ*(

_{n}*ρ*)/

*dρ*is the logarithmic derivative of the Riccati-Bessel function

*ψ*.

_{n}*a*=

_{n}*b*if the relative impedance

_{n}*η*=1. From Eqs. (4.74) in Ref. [16], the complex amplitude scattering matrix elements can be written as:

*θ*=180° since

*π*(180°)+

_{n}*τ*(180°)=0 for all

_{n}*n*. The radar backscattering cross section

*σ*is defined as the area which is 4

_{b}*π*times the differential scattering cross section for scattering into a unit solid angle around the backscattering direction [16]. It is then evident that

*σ*is equal to zero for

_{b}*η*=1. For large absorbing spherical particles, the asymptotic expression of the radar back scattering efficiency

*Q*=

_{b}*σ*/

_{b}*π*

*a*

^{2}has a value of (Eq. 4.83 in Ref. [16])

*R*(0°) is the reflectance at normal incidence at the surface of the sphere.

*R*(0°) vanishes if the impedance matching is satisfied, namely,

*η*=1 (see pp. 306, Ref. [17]).

*S*

_{12}=

*S*

_{34}=0 and

*S*

*11*=

*S*

_{33}for particles with

*η*=1. Only diagonal elements survive for this case.

*μ*=1 is then assumed for the particle and surrounding medium to obtain

*a*and

_{n}*b*. To reinstate the dependence of

_{n}*μ*in

*a*and

_{n}*b*, the following definitions are convenient:

_{n}*a*and

*b*are the inner and outer radii of the coated sphere, respectively; m

_{i},

*i*=1,2 are the relative refractive indices in the region

*i*; the subscripts

*i*=1,2 denote that the quantities are for the regions of 0<

*r*<

*a*and

*a*<

*r*<

*b*, respectively;

*ε*3 and

*μ*3 are the permittivity and permeability of the surrounding medium, respectively. The expansion coefficients of the scattered field can be solved from Eqs. (8.1) in Ref. [16]; given by:

*χ*(

_{n}*ρ*)=-

*ρy*(

_{n}*ρ*) is used in Eqs. (11). If

*η*

_{1}=

*η*

_{2}=1, it is obvious that

*A*=

_{n}*B*,

_{n}*D̃*=

_{n}*G̃*, and

_{n}*a*=

_{n}*b*. According to Eqs. (8), it is the same as the homogeneous sphere since

_{n}*S*

_{11}(

*θ*=180°)=0. Furthermore,

*S*

_{12}=

*S*

_{34}=0 and

*S*

_{11}=

*S*

_{33}also hold for coated spheres if

*η*

_{1}=

*η*

_{2}.

## 3. Numerical results

19. W.J. Wiscombe, “Improved Mie scattering algorithms,” Appl. Opt. **19**, 1505–1509 (1980). [CrossRef] [PubMed]

20. O. B. Toon and T. P. Ackerman, “Algorithms for the calculation of scattering by starified spheres,” Appl. Opt. **20**, 3657–3660 (1981). [CrossRef] [PubMed]

*η*=1, it would be interesting to show how fast

*S*

_{11}approaches zero around

*θ*=180° for different values of m. Figure 1 plots the phase function, the normalized scattering matrix element

*S*

_{11}, for m=1.1,1.3,1.7,1.1+0.1i,1.3+0.1i, and 1.7+0.1i. In this computation,

*η*=1 and the size parameter

*x*=10 are assumed. Evidently, the phase function approaches zero faster in the region of 170°<

*θ*<180° as the relative refractive index decreases. For m=1.3 and 1.7, the rates of decrease of the phase functions are almost the same. The rate of decrease of the phase functions stops if the refractive index reaches a certain value. In addition, an imaginary part of the refractive index causes the phase function to decrease faster. Table 1 shows the extinction efficiencies

*Q*and scattering efficiencies

_{ext}*Q*for the cases in Fig. 1. The cases with zero imaginary part of the refractive indices, the extinction and scattering efficiencies are equal. It is interesting that the extinction efficiency for m=1.3+0.1i is smaller than that of m=1.3, while the extinction efficiencies of m=1.1+0.1i and m=1.7+0.1i are larger than those of m=1.1 and m=1.7, respectively.

_{sca}*η*, Fig. 2 shows the phase functions for spherical particles with

*η*=0.9999,0.999,0.99,0.91for scattering angles ranging from 160° to 180°. A size parameter of

*x*=10 and a relative refractive index of m=1.3 are used. It is shown that the backscatter is small if

*η*remains close to 1. As

*η*deviates from 1, the backscatter also gradually deviates from 0. Table 2 shows the extinction/scattering efficiencies for cases in Fig. 2. The extinction and scattering efficiencies are equal for the four cases because the imaginary part of the refractive indices are zero. The extinction efficiencies do not have significant changes until

*η*reaches 0.91.

*k*is the wave vector in the surrounding medium;

*x*=

*ka*and

*y*=

*kb*are the size parameters of the inner and outer spheres, respectively; Im(m

_{2}) is the imaginary part of the relative refractive index of the outer layer.

_{2}=2+0.5i and m

_{2}=2+i.

*η*

_{2}=1 is used to insure the outer layer produces a zero backscatter. The size parameters for the inner sphere is

*x*=5. A refractive index of m

_{1}=1.5 is assumed for the inner core to test its sensitivity. The impedance for the inner sphere is

*η*

_{1}=2/3. Thus the permeability of the inner core is taken to be 1. We noticed that τ=(

*y*-

*x*) Im(m

_{2}) is the critical factor to achieve a zero backscatter. In Fig. 3, two values of

*τ*=1 and 4 are chosen to test the sensitivity. For each value of

*τ*and Im(m

_{2}), the corresponding value of

*y*can be determined by

*τ*=(

*y*-

*x*)Im(m

_{2}). The phase function for the inner sphere without the coated sphere is also shown for comparison. It is evident from Fig. 3 that the coated sphere decreases the phase function at the backscatter. The phase functions for τ=4 have negligible values while the phase functions for τ=1 still have significant values at the backscatter. This feature confirms that τ is the dominant factor that determines the magnitude of the backscatter. This is a remarkable result because τ=

*k*(

*b*-

*a*)Im(m

_{2})=4 is just a fairly moderate value and the backscatter has already been reduced by several orders of magnitude. Table 3 shows the extinction and scattering efficiencies for cases in Fig. 3. The particles shown in Fig. 3 all have different sizes, which makes it hard to interpret the efficiencies. To overcome this problem, the ratio of the extinction/scattering cross sections for the coated particle and the “Not coated” particle is introduced.

*C*is the extinction/scattering cross section for the “Not coated” case;

^{Nc}_{ext,sca}*C*is the extinction/scattering cross section for the coated particle. The proportionality between the size parameter and the radius of a spherical particle is used to get the right hand side of Eq. (13). The size parameter

_{ext,sca}*x*of the “Not coated” case is equal to that of the cores of the coated particles. Equation (13) represents the ratio of the power eliminated/scattered from the incident light for the coated and “Not coated” particles. Table 3 shows the extinction cross sections for the coated particles generally larger than that of the “Not coated” particle. The only exception is in the case of τ=1 and Im(m

_{2})=0.5. It means that the negligible backscatters are achieved at the cost of larger extinction cross sections.

## 4. Conclusions

*η*

_{2}=1 holds for its outer layer and τ=(

*y*-

*x*)Im(m

_{2}) is sufficiently large. In other words, this coated sphere can act as an invisible cloak for monostatic lidar/radar detection. Furthermore, the extinction cross section for the coated sphere is generally larger than the “Not coated” particle.

## Acknowledgments

## 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. | 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. | A. J. Ward and J. B. Pendry, “Refraction and geometry in maxwells equations,” J. Mod. Opt. |

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

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

7. | F. Zolla, S. Guenneau, A. Nicolet, and J. B. Pendry, “Electromagnetic analysis of cylindrical invisibility cloaks and the mirage effect,” Opt. Lett. |

8. | H. Chen, B. I. Wu, B. Zhang, and J. A. Kong, “Electromagnetic Wave Interactions with a Metamaterial Cloak,” Phys. Rev. Lett. |

9. | B. Zhang, H. Chen, B. Wu, Y. Luo, L. Ran, and J. A. Kong, “Response of a cylindrical invisibility cloak to electromagnetic waves,” Phys. Rev. B |

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

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

12. | G. Mie, “Beigrade zur optik truber medien, speziell kolloidaler metallosungen,” Ann. Phys. (Leipzig) |

13. | J. Maxwell Garnett, “Colours in metal glasses and in metallic films,” Philos. Trans. R. Soc. London |

14. | J.R. Liu, M. Itoh, and K.-I Machida, “Frequency dispersion of complex permeability and permittivity on ironbased nanocomposites derived from rare earth-iron intermetallic compounds,” J. Alloys Compd. |

15. | H.C. van de Hulst, |

16. | C. F. Bohren and D. R. Huffman, |

17. | J. D. Jackson, |

18. | W. J. Wiscombe, “Mie scattering calculation,” NCAR Tech. Note TN-140+STR (National Center for Atmospheric Research, Boulder, Colo., 1979). |

19. | W.J. Wiscombe, “Improved Mie scattering algorithms,” Appl. Opt. |

20. | O. B. Toon and T. P. Ackerman, “Algorithms for the calculation of scattering by starified spheres,” Appl. Opt. |

21. | P. -W. Zhai, Y. -K. Lee, G. W. Kattawar, and P. Yang, “Implementing the Near- to Far-Field Transformation in the Finite-Difference Time-Domain Method,” Appl. Opt. |

**OCIS Codes**

(290.5850) Scattering : Scattering, particles

(280.1350) Remote sensing and sensors : Backscattering

**ToC Category:**

Scattering

**History**

Original Manuscript: October 17, 2007

Revised Manuscript: January 15, 2008

Manuscript Accepted: January 17, 2008

Published: January 18, 2008

**Citation**

Peng-Wang Zhai, Yu You, George W. Kattawar, and Ping Yang, "Monostatic lidar/radar invisibility using coated spheres," Opt. Express **16**, 1431-1439 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-3-1431

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### References

- J. B. Pendry, D. Schurig, and D. R. Smith, "Controlling Electromagnetic Fields," Science 312, 1780-1782 (2006). [CrossRef] [PubMed]
- U. Leonhardt, "Optical Conformal Mapping," Science 312, 1777-1780 (2006). [CrossRef] [PubMed]
- 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]
- A. J. Ward and J. B. Pendry,"Refraction and geometry in maxwells equations," J. Mod. Opt. 43, 773-793 (1996). [CrossRef]
- 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]
- S. A. Cummer, B.-I. Popa, D. Schurig, D. R. Smith, and J. B. Pendry, "Full-wave simulations of electromagnetic cloaking structures," Phys. Rev. E 74, 036621-1-5 (2006).
- F. Zolla, S. Guenneau, A. Nicolet, and J. B. Pendry, "Electromagnetic analysis of cylindrical invisibility cloaks and the mirage effect," Opt. Lett. 32, 1069-1071 (2007). [CrossRef] [PubMed]
- H. Chen, B. I. Wu, B. Zhang, and J. A. Kong, "Electromagnetic Wave Interactions with a Metamaterial Cloak," Phys. Rev. Lett. 99, 063903-1-4 (2007). [CrossRef] [PubMed]
- B. Zhang, H. Chen, B. Wu, Y. Luo, L. Ran, and J. A. Kong, "Response of a cylindrical invisibility cloak to electromagnetic waves," Phys. Rev. B 76, 121101-1-4 (R) (2007) [CrossRef]
- A. Alu and N. Engheta, "Achieving transparency with plasmonic and metamaterial coatings," Phys. Rev. E 72, 016623-1-9 (2005). [CrossRef]
- G. W. Milton and N.-A. P. Nicorovici, "On the cloaking effects associated with anomalous localized resonance," Proc. R. Soc. London, Ser. A 462, 3027-3059 (2006). [CrossRef]
- G. Mie, "Beigrade zur optik truber medien, speziell kolloidaler metallosungen," Ann. Phys. (Leipzig) 25, 377- 455 (1908). [CrossRef]
- J. Maxwell Garnett, "Colours in metal glasses and in metallic films," Philos. Trans. R. Soc. London 203, 385 (1904). [CrossRef]
- J.R. Liu, M. Itoh, and K.-I Machida, "Frequency dispersion of complex permeability and permittivity on ironbased nanocomposites derived from rare earth-iron intermetallic compounds," J. Alloys Compd. 408-412, 1396- 1399 (2006). [CrossRef]
- H.C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).
- C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
- J. D. Jackson, Classical Electrodynamics, 3rd Edition (Wiley-VCH, 1998),
- W. J. Wiscombe, "Mie scattering calculation," NCAR Tech. Note TN-140+STR (National Center for Atmospheric Research, Boulder, Colo., 1979).
- W.J. Wiscombe, "Improved Mie scattering algorithms," Appl. Opt. 19, 1505-1509 (1980). [CrossRef] [PubMed]
- O. B. Toon and T. P. Ackerman, "Algorithms for the calculation of scattering by starified spheres," Appl. Opt. 20, 3657-3660 (1981). [CrossRef] [PubMed]
- P. -W. Zhai, Y. -K. Lee, G. W. Kattawar, and P. Yang, "Implementing the Near- to Far-Field Transformation in the Finite-Difference Time-Domain Method," Appl. Opt. 43, 3738-3746 (2004) [CrossRef] [PubMed]

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