## Enhancing and suppressing radiation with some permeability-near-zero structures |

Optics Express, Vol. 18, Issue 16, pp. 16587-16593 (2010)

http://dx.doi.org/10.1364/OE.18.016587

Acrobat PDF (1040 KB)

### Abstract

Using some special properties of a permeability-near-zero material, the radiation of a line current is greatly enhanced by choosing appropriately the dimension of a dielectric domain in which the source lies and that of a permeability-near-zero shell. The radiation of the source can also be completely suppressed by adding appropriately another dielectric domain or an arbitrary perfect electric conductor (PEC) inside the shell. Enhanced directive radiation is also demonstrated by adding a PEC substrate.

© 2010 OSA

## 1. Introduction

1. J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs I, “Extremely low frequency plasmons in metallic mesostructures,” Phys. Rev. Lett. **76**(25), 4773–4776 (1996). [CrossRef] [PubMed]

2. J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors, and enhanced non-linear phenomena,” IEEE Trans. Microw. Theory Tech. **47**(11), 2075–2084 (1999). [CrossRef]

3. D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. **84**(18), 4184–4187 (2000). [CrossRef] [PubMed]

4. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. **85**(18), 3966–3969 (2000). [CrossRef] [PubMed]

7. 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**(5801), 977–980 (2006). [CrossRef] [PubMed]

8. R. W. Ziolkowski, “Propagation in and scattering from a matched metamaterial having a zero index of refraction,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **70**(4 Pt 2), 046608 (2004). [CrossRef] [PubMed]

17. M. G. Silveirinha and N. Engheta, “Transporting an Image through a Subwavelength Hole,” Phys. Rev. Lett. **102**(10), 103902 (2009). [CrossRef] [PubMed]

9. S. Enoch, G. Tayeb, P. Sabouroux, N. Guérin, and P. Vincent, “A metamaterial for directive emission,” Phys. Rev. Lett. **89**(21), 213902 (2002). [CrossRef] [PubMed]

11. Y. G. Ma, P. Wang, X. Chen, and C. K. Ong, “Near-field plane-wave-like beam emitting antenna fabricated by anisotropic metamaterial,” Appl. Phys. Lett. **94**(4), 044107 (2009). [CrossRef]

12. M. Silveirinha and N. Engheta, “Tunneling of electromagnetic energy through subwavelength channels and bends using *ε*-near-zero materials,” Phys. Rev. Lett. **97**(15), 157403 (2006). [CrossRef] [PubMed]

15. R. P. Liu, Q. Cheng, T. Hand, J. J. Mock, T. J. Cui, S. A. Cummer, and D. R. Smith, “Experimental demonstration of electromagnetic tunneling through an epsilon-near-zero metamaterial at microwave frequencies,” Phys. Rev. Lett. **100**(2), 023903 (2008). [CrossRef] [PubMed]

16. A. Alù, M. G. Silveirinha, A. Salandrino, and N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: Tailoring the radiation phase pattern,” Phys. Rev. B **75**(15), 155410 (2007). [CrossRef]

17. M. G. Silveirinha and N. Engheta, “Transporting an Image through a Subwavelength Hole,” Phys. Rev. Lett. **102**(10), 103902 (2009). [CrossRef] [PubMed]

## 2. Enhancing radiation

*x*-

*y*plane, and the time harmonic factor is

*exp*(−

*iωt*). The structure shown in Fig. 1(a) is investigated first. Domain 4 is not considered (i.e., domain 4 vanishes) for the moment. Domain 3 is surrounded by domain 2 of relative permeability

*μ*

_{2}as the shell, and domain 1 is the background. Domains 1 and 3 are of common dielectric. A unit line current propagating along the

*z*axis is inside domain 3 as an excitation source. Let

*F*(

_{n}**r**) denote some quantity

*F*(

**r**) in domain

*n*(

*n*= 1,2,3). In domain 2, the electromagnetic field satisfies Maxwell’s equations, where

*ε*

_{0}and

*μ*

_{0}are the vacuum permittivity and permeability, respectively, and subscripts

*x*,

*y*, and

*z*represent a component of an electromagnetic quantity along some canonical base. One can rewrite Eq. (2) for the shell region in the following integration form,where integration contours

*∂*2 and

*∂*3 are shown in Fig. 1(a), and

**H**

_{2,}

*(*

_{∂n}**r**) denotes the value of magnetic field

**H**

_{2}(

**r**) at boundary

*∂n*. When

*μ*

_{2}tends to zero, electric field

*E*

_{2,}

*(*

_{z}**r**) in domain 2 is uniform and assumed to be constant

*E*. Then, the right side of Eq. (3) can be written as −

_{z}*iωε*

_{0}

*S*

_{2}

*E*where

_{z}*S*

_{2}is the area of domain 2. According to the continuity condition of the tangential electric and magnetic fields at both sides of a boundary [18],

*E*

_{1,}

_{z}_{,}

_{∂}_{2}(

**r**) and

*E*

_{3,}

_{z}_{,}

_{∂}_{3}(

**r**) are also equal to

*E*, and the contour integrals of

_{z}**H**

_{2,}

*(*

_{∂n}**r**) at the left side of Eq. (2) can be replaced by those of

**H**

_{1,}

_{∂}_{2}(

**r**) and

**H**

_{3,}

_{∂}_{3}(

**r**), respectively. Following the unique theorem [18], if one knows the tangential electric field at the boundary of a domain and the excitation source in the domain, the electromagnetic field in the whole domain is uniquely determined. Therefore, both sides of Eq. (3), as well as the fields in domains 1 and 3, are independent of the position of domain 3 inside domain 2.

*r*

_{2}and

*r*

_{3}, respectively, and the unit line current is at the center of domain 3, the value of

*E*can be analytically obtained. The electromagnetic field in each domain can be expanded by the Bessel and Hankel functions [18]. Since the electric field on boundary

_{z}*∂*3 is equal to constant

*E*everywhere, and the line current is at the center of domain 3, the electric field in domain 3 is only composed of the zero-order Bessel and Hankel functions,where

_{z}*k*

_{3}is the wave-number in domain 3. Similarly, the electric field in domain 1 is only composed of the zero-order Bessel function,

*a*

_{1}and

*a*

_{3}can be expressed as functions of

*E*based on the boundary condition that both

_{z}*E*

_{1,}

_{z}_{,}

_{∂}_{2}(

**r**) and

*E*

_{3,}

_{z}_{,}

_{∂}_{3}(

**r**) are equal to

*E*. When electric field

_{z}*E*

_{n}_{,}

*(*

_{z}**r**) in domain

*n*is known, the corresponding magnetic field

**H**

*(*

_{n}**r**) can be obtained from Maxwell’s equations. Then, according to Eq. (3), one obtains the value of

*E*,

_{z}*P*, i.e.,

_{norm}*F*). If |

*F*| can be small, |

*E*| and

_{z}*P*will be large. There are three items in the expression of

_{norm}*F*whose real parts can cancel each other. When domains 1 and 3 are lossless, since there is only one complex item in the expression of

*F*, imag(

*F*) cannot be zero, unlike real(

*F*). However, if |imag(

*F*)| can be small, large |

*E*| and

_{z}*P*can still be obtained. This condition can be fulfilled as shown in the following example. Domains 1 and 3 are assumed to be of free space. When

_{norm}*r*

_{2}= 2

*λ*

_{0}(

*λ*

_{0}is the wavelength in free space), Fig. 2 shows the real and imaginary parts of

*F*as well as the corresponding

*P*as

_{norm}*r*

_{3}varies. As shown in Fig. 2(a), real(

*F*) is zero at some special values of

*r*

_{3}. The appearance of several zero roots of real(

*F*) is because the Bessel functions are oscillating functions. As shown in Fig. 2(b),

*P*is locally maximal at these special values of

_{norm}*r*

_{3}, and the radiation is greatly enhanced. When

*r*

_{3}= 0.38775

*λ*

_{0},

*P*is about 190. The whole structure behaves as a resonator to enhance the radiation. Because real(

_{norm}*F*) steeply passes through the zero points as shown in Fig. 2(a), the peaks of

*P*in Fig. 2(b) are very narrow. This indicates that the enhanced radiation is sensitive to the dimension of domain 3. In the inset of Fig. 2(b), one can see that to keep

_{norm}*P*in the same order of magnitude and strongly enhance the radiation, three figures are necessary for

_{norm}*r*

_{3}. The radiation can also be influenced by the dimension of domain 2, which can be stronger enhanced as

*r*

_{2}increases. This may be in contrast to our common intuition. Based on the fact that the normal transmission through a permeability-zero slab is approximately inversely proportional to the thickness [10

10. Y. Jin and S. L. He, “Impedance-matched multilayered structure containing a zero-permittivity material for spatial filtering,” J. Nonlinear Opt. Phys. Mater. **17**(03), 349–355 (2008). [CrossRef]

## 3. Completely suppressing radiation

*r*

_{4}inside domain 2. Figure 3 shows

*P*as a function of radius

_{norm}*r*

_{4}when domain 4 is circular of free space,

*r*

_{2}= 2

*λ*

_{0}, and

*r*

_{3}= 0.38775

*λ*

_{0}. The existence of domain 4 greatly influences the radiation. When domain 4 is large,

*P*is very large only in a narrow range of

_{norm}*r*

_{4}. When

*r*

_{4}is at some special values,

*P*is zero. This can be understood in the following way. The electric field inside domain 4 is just composed of the zero-order Bessel function,

_{norm}*a*

_{4}

*J*

_{0}(

*k*

_{4}

*r*

_{4}), and

*E*can still be expressed by Eq. (6) after adding in the denominator a term of −

_{z}*k*

_{4}

*r*

_{4}

*J*

_{1}(

*k*

_{4}

*r*

_{4})/

*J*

_{0}(

*k*

_{4}

*r*

_{4}).

*J*

_{0}(

*x*) is an oscillating function with zero roots. When

*k*

_{4}

*r*

_{4}is a zero root of

*J*

_{0}(

*x*), electric field

*E*

_{4,}

_{z}_{,}

_{∂}_{4}(

**r**) is zero on boundary

*∂*4, and consequently

*E*and

_{z}*E*

_{1,}

_{z}_{,}

_{∂}_{2}(

**r**) are forced to be zero. This means that no power can be radiated into the background, i.e., the radiation is completely suppressed. This is a rather interesting result. Just by adding a common dielectric material, permeability-zero domain 2 behaves like a PEC, inside which the electric field is zero. If the excitation source is moved outside domain 2, the radiated wave seems scattered by a PEC possessing the shape of domain 2. However, domain 2 is not completely like a PEC, as the internal magnetic field is not zero, which will be shown below.

## 4. Numerical simulations

19. Comsol Multiphysics 3.4, Comsol Inc., www.comsol.com.

*μ*

_{2}= 10

^{−5}+ 10

^{−4}

*i*, which deviates a bit from zero. The width and height of domain 2 are

*w*

_{2}= 8

*λ*

_{0}and

*h*

_{2}=

*λ*

_{0}, respectively. The centers of domains 2 and 3 are at the same point with the line current. When the width and height of domain 3 are appropriately chosen, the radiation of the line current can be strongly enhanced. Such an example is

*w*

_{3}= 0.8

*λ*

_{0}and

*h*

_{3}= 0.66

*λ*

_{0}, for which

*P*is 46. Figure 4(a) shows a snapshot of the electric field distribution and Fig. 4(b) shows the magnetic amplitude distribution. The electromagnetic wave is radiated in all the directions, and most energy flows nearly normally through the long top and bottom surfaces of domain 2.

_{norm}*k*= 3

_{x}*k*/2,

_{y}*m*= 1, and

*n*= 3, then one can get the corresponding values of

*w*

_{4}and

*h*

_{4}according to Eq. (8). Figure 4(c) shows a snapshot of the electric field distribution, and Fig. 4(d) shows the magnetic amplitude distribution. The electric field in domains 1 and 2 is very weak, but not exactly zero because

*μ*

_{2}deviates a bit from zero, and the radiation is also very weak with

*P*only 0.1. The magnetic field in domain 2 is nonzero and rather non-uniform.

_{norm}*P*as a function of the distance (

_{norm}*d*) between domains 2 and 5. When

*d*is small,

*P*is small (as expected). When

_{norm}*d*increases gradually,

*P*also increases. When

_{norm}*d*= 0.38

*λ*

_{0}, the radiation of the line current is most strongly enhanced, and

*P*reaches 111, which is larger than that when domain 5 is not added. This indicates that when

_{norm}*d*is appropriately chosen, the reflected wave by domain 5 can coherently help enhance the radiation. Figure 5(b) shows a snapshot of the electric field distribution, and Fig. 5(c) shows the magnetic amplitude distribution, in which one can see clearly that the upward propagation of the radiated electromagnetic field is very directive.

## 5. Conclusion

## Acknowledgments

## References and links

1. | J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs I, “Extremely low frequency plasmons in metallic mesostructures,” Phys. Rev. Lett. |

2. | J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors, and enhanced non-linear phenomena,” IEEE Trans. Microw. Theory Tech. |

3. | D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. |

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

5. | N. Fang and X. Zhang, “Imaging properties of a metamaterial superlens,” Appl. Phys. Lett. |

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

7. | 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 |

8. | R. W. Ziolkowski, “Propagation in and scattering from a matched metamaterial having a zero index of refraction,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

9. | S. Enoch, G. Tayeb, P. Sabouroux, N. Guérin, and P. Vincent, “A metamaterial for directive emission,” Phys. Rev. Lett. |

10. | Y. Jin and S. L. He, “Impedance-matched multilayered structure containing a zero-permittivity material for spatial filtering,” J. Nonlinear Opt. Phys. Mater. |

11. | Y. G. Ma, P. Wang, X. Chen, and C. K. Ong, “Near-field plane-wave-like beam emitting antenna fabricated by anisotropic metamaterial,” Appl. Phys. Lett. |

12. | M. Silveirinha and N. Engheta, “Tunneling of electromagnetic energy through subwavelength channels and bends using |

13. | Y. Jin, P. Zhang, and S. L. He, “Squeezing electromagnetic energy with a dielectric split ring inside a permeability-near-zero metamaterial,” Phys. Rev. B |

14. | B. Edwards, A. Alù, M. E. Young, M. Silveirinha, and N. Engheta, “Experimental verification of epsilon-near-zero metamaterial coupling and energy squeezing using a microwave waveguide,” Phys. Rev. Lett. |

15. | R. P. Liu, Q. Cheng, T. Hand, J. J. Mock, T. J. Cui, S. A. Cummer, and D. R. Smith, “Experimental demonstration of electromagnetic tunneling through an epsilon-near-zero metamaterial at microwave frequencies,” Phys. Rev. Lett. |

16. | A. Alù, M. G. Silveirinha, A. Salandrino, and N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: Tailoring the radiation phase pattern,” Phys. Rev. B |

17. | M. G. Silveirinha and N. Engheta, “Transporting an Image through a Subwavelength Hole,” Phys. Rev. Lett. |

18. | J. A. Stratton, |

19. | Comsol Multiphysics 3.4, Comsol Inc., www.comsol.com. |

**OCIS Codes**

(260.0260) Physical optics : Physical optics

(350.5610) Other areas of optics : Radiation

(160.3918) Materials : Metamaterials

**ToC Category:**

Physical Optics

**History**

Original Manuscript: June 14, 2010

Revised Manuscript: July 17, 2010

Manuscript Accepted: July 19, 2010

Published: July 22, 2010

**Citation**

Yi Jin and Sailing He, "Enhancing and suppressing radiation with some permeability-near-zero structures," Opt. Express **18**, 16587-16593 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-16-16587

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### References

- J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely low frequency plasmons in metallic mesostructures,” Phys. Rev. Lett. 76(25), 4773–4776 (1996). [CrossRef] [PubMed]
- J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors, and enhanced non-linear phenomena,” IEEE Trans. Microw. Theory Tech. 47(11), 2075–2084 (1999). [CrossRef]
- D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84(18), 4184–4187 (2000). [CrossRef] [PubMed]
- J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000). [CrossRef] [PubMed]
- N. Fang and X. Zhang, “Imaging properties of a metamaterial superlens,” Appl. Phys. Lett. 82(2), 161–163 (2003). [CrossRef]
- J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (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(5801), 977–980 (2006). [CrossRef] [PubMed]
- R. W. Ziolkowski, “Propagation in and scattering from a matched metamaterial having a zero index of refraction,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(4 Pt 2), 046608 (2004). [CrossRef] [PubMed]
- S. Enoch, G. Tayeb, P. Sabouroux, N. Guérin, and P. Vincent, “A metamaterial for directive emission,” Phys. Rev. Lett. 89(21), 213902 (2002). [CrossRef] [PubMed]
- Y. Jin and S. L. He, “Impedance-matched multilayered structure containing a zero-permittivity material for spatial filtering,” J. Nonlinear Opt. Phys. Mater. 17(03), 349–355 (2008). [CrossRef]
- Y. G. Ma, P. Wang, X. Chen, and C. K. Ong, “Near-field plane-wave-like beam emitting antenna fabricated by anisotropic metamaterial,” Appl. Phys. Lett. 94(4), 044107 (2009). [CrossRef]
- M. Silveirinha and N. Engheta, “Tunneling of electromagnetic energy through subwavelength channels and bends using ε-near-zero materials,” Phys. Rev. Lett. 97(15), 157403 (2006). [CrossRef] [PubMed]
- Y. Jin, P. Zhang, and S. L. He, “Squeezing electromagnetic energy with a dielectric split ring inside a permeability-near-zero metamaterial,” Phys. Rev. B 81(8), 085117 (2010). [CrossRef]
- B. Edwards, A. Alù, M. E. Young, M. Silveirinha, and N. Engheta, “Experimental verification of epsilon-near-zero metamaterial coupling and energy squeezing using a microwave waveguide,” Phys. Rev. Lett. 100(3), 033903 (2008). [CrossRef] [PubMed]
- R. P. Liu, Q. Cheng, T. Hand, J. J. Mock, T. J. Cui, S. A. Cummer, and D. R. Smith, “Experimental demonstration of electromagnetic tunneling through an epsilon-near-zero metamaterial at microwave frequencies,” Phys. Rev. Lett. 100(2), 023903 (2008). [CrossRef] [PubMed]
- A. Alù, M. G. Silveirinha, A. Salandrino, and N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: Tailoring the radiation phase pattern,” Phys. Rev. B 75(15), 155410 (2007). [CrossRef]
- M. G. Silveirinha and N. Engheta, “Transporting an Image through a Subwavelength Hole,” Phys. Rev. Lett. 102(10), 103902 (2009). [CrossRef] [PubMed]
- J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).
- Comsol Multiphysics 3.4, Comsol Inc., www.comsol.com .

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