## A cone-shaped concentrator with varying performances of concentrating

Optics Express, Vol. 16, Issue 10, pp. 6809-6814 (2008)

http://dx.doi.org/10.1364/OE.16.006809

Acrobat PDF (267 KB)

### Abstract

A three-dimensional cone-shaped concentrator was designed and analyzed through an approach of coordinate transformation theory. The device can provide varying performances for concentrating along the symmetric axis. The physical picture regarding concentrating ability of this structure was revealed and quantitative analyses were performed for the purpose of investigating the dependence of the concentrating properties on the structural parameters. Moreover, reduced material parameters were theoretically derived and the corresponding mismatched impedance at boundaries was analyzed. Finite element method-based numerical simulations results of the device were further presented to verify our theoretical design.

© 2008 Optical Society of America

## 1. Introduction

*et al.*, [1

1. A. J. Ward and J. B. Pendry, “Refraction and geometry in Maxwell’s equation,” J. Mod. Opt. **43**, 773–793 (1996). [CrossRef]

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

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 **314**, 977–980 (2006). [CrossRef] [PubMed]

4. M. Tsang and D. Psaltis, “Magnifying perfect lens and superlens design by coordinate transformation,” http://www.arxiv.org:physics/0708.0262v1, (2007).

5. 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]

6. H. Chen and C. T. Chan, “Transformation media that rotate electromagnetic fields,” Appl. Phys. Lett , **90**, 241105 (2007). [CrossRef]

7. M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, and D. R. Smith, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell’s Equations,” http://www.arxiv.org:physics/0706.2452v1, (2007).

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 **314**, 977–980 (2006). [CrossRef] [PubMed]

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

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 **314**, 977–980 (2006). [CrossRef] [PubMed]

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

7. M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, and D. R. Smith, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell’s Equations,” http://www.arxiv.org:physics/0706.2452v1, (2007).

## 2. Principle

7. M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, and D. R. Smith, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell’s Equations,” http://www.arxiv.org:physics/0706.2452v1, (2007).

*R*

_{2}) into a circle with a radius of

*R*

_{1}, named as region I hereinafter; the second one is expanding the space between

*R*and

_{2}*R*to a region between

_{3}*R*and

_{1}*R*so as to make the space continuous, named region II hereinafter. Figure 1(b) is the schematic diagram of the space transformation. Note that

_{3}*R*(

_{i}*i*=1, 2, 3) is a function of

*z*in this device. The process can be mathematically described as:

1. A. J. Ward and J. B. Pendry, “Refraction and geometry in Maxwell’s equation,” J. Mod. Opt. **43**, 773–793 (1996). [CrossRef]

*R*=

_{1}*R*-

_{10}*zt*,

*R*=

_{2}*R*-

_{20}*zt*,

*R*=

_{3}*R*-

_{30}*zt*, where

*z*is the symmetric axis,

*R*(

_{i0}*i*=1, 2, 3) is the radius at

*z*=0, and

*t*is the slope of the cone. The ratio

*R*/

_{2}*R*that affects the concentrating performance greatly can be expressed as

_{1}*r*), azimuthally (

*φ*) and

*z*directions which are hard to satisfy at the same time. Here a TE mode design is considered and the TM mode design follows the same principle by making

*ε*→

*µ*and

*µ*→

*ε*substitutions. In the TE mode design case, only

*ε*,

_{z}*µ*and

_{φ}*µ*enter into the Maxwell’s equations. Therefore, we just consider the

_{r}*ε*,

_{z}*µ*and

_{φ}*µ*, and assume that the other three parameters,

_{r}*µ*,

_{z}*ε*and

_{φ}*ε*, are units which means the TM mode energy will remain the same as in free space. The reduced parameters that make the same solutions for the wave equations are obtained as long as the product of

_{r}*µ*and

_{r}ε_{z}*µ*remain the same value [3

_{φ}ε_{z}**314**, 977–980 (2006). [CrossRef] [PubMed]

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

*ε*and

_{z}*µ*are constants, only

_{φ}*µ*is dependent on the coordinate; and there are numerous sets of the parameters, theoretically. Considering the convenience of fabrication and the matching condition at the boundary, three sets of the reduced material parameters are derived. For the first set,

_{r}*ε*=1, which is easy to realize; for the second set, the parameters in

_{z}*φ*and

*z*directions are the same, and the impedance matches at the boundary of

*r*=

*R*; and the third set has an impedance matched at the boundary of

_{3}*r*=

*R*

_{1}. They can be written in the following equations respectively.

## 3. Ideal material parameters of the cone-shaped concentrator

*z*axis is the symmetric axis. The incident wave is TE polarized with the frequency of 4GHz, which propagates along the y axis with parameters of

*R*=1.5 cm,

_{10}*R*=3.5 cm,

_{20}*R*=5.5 cm and

_{30}*t*=1/15. Figure 2(a) shows the distribution of

*E*in

_{z}*x*-

*y*plane and

*y*-

*z*plane respectively. And the distribution of

*E*in a cross-section of the cone parallel to the

_{z}*x*-

*y*plane at

*z*=6 cm with

*R*=1.1 cm,

_{1}*R*=3.1 cm and

_{2}*R*=5.1 cm is presented in Fig. 2(b). The electric field is concentrated to the region with a radius of

_{3}*R*. The wavelength in free space is about

_{1}*R*/

_{2}*R*times of that in region I.

_{1}*r*will lead to a larger wave number and a smaller velocity.

*R*/

_{2}*R*, because we have compressed the space with radius of

_{1}*R*into a region with radius of

_{2}*R*in the first step of the transformation. Figure 3(a) shows the energy distribution along the

_{1}*z*axis at

*x*=

*y*=0. It can be seen that the energy varies with different position of

*z*, which means that the ability of concentrating is not constant along the symmetrical axis. Figure 3(b) represents the relationship between

*R*/

_{2}*R*and the concentrating ability. The energy distribution in free space is marked by the red line, and the average value along the line is 4.5×10

_{1}^{-12}J/m

^{2}. The green curve represent energy distribution along the

*y*axis of the concentrator at site of

*x*=0 and

*z*=7.5 cm with

*R*=1 cm,

_{1}*R*=3 cm and

_{2}*R*=5 cm, so

_{3}*R*/

_{2}*R*=3. Obviously, the energy in the center, whose average value is 4.25×10

_{1}^{-11}J/m

^{2}, is about 9 times larger than that of the value in free space. A similar result can be obtained from the yellow curve which represents energy distribution along

*y*axis at

*x*=0 and

*z*=0 with

*R*=1.5 cm,

_{1}*R*=3.5 cm and

_{2}*R*=5.5 cm. In this case

_{3}*R*/

_{2}*R*=2.3, the averaged energy value in region I is 2.4×10e

_{1}^{-11}J/m

^{2}, so the enhancement obtained from Fig. 3(b) is ~5.3. The averaged energy in region I of the blue line at

*x*=0 and

*z*=-7.5 cm with

*R*=2 cm,

_{1}*R*=4 cm and

_{2}*R*=6 cm is 1.68×10

_{3}^{-11}J/m

^{2}which is almost 4 times larger than the case in free space. The coarseness of the lines is caused by the finite number from the finite elements.

## 4. Reduced parameters of the cone-shaped concentrator

*µ*and

_{r}ε_{z}*µ*to be the same, three sets of reduced parameters have been derived in section2 for their specificity. Figure 4 shows the impedance of four conditions at the site of

_{φ}ε_{z}*x*=

*z*=0. For the ideal case, the impedance is

*Z*=(

*R*-

_{3}*R*)/[(

_{1}*R*-

_{3}*R*)

_{2}*k*], which is matched at both boundaries with

*r*=

*R*and

_{3}*r*=

*R*. The impedance of the reduced parameters is

_{1}*Z*=(

*R*-

_{3}*R*)/(

_{2}*R*-

_{3}*R*) for set 1 (green line), and it does not match the impedance at both boundaries of

_{1}*r*=

*R*and

_{1}*r*=

*R*in this case; it can be matched at the inner boundary at

_{3}*r*=

*R*only in the case of

_{1}*R*=

_{3}*R*+

_{2}*R*. The impedance is

_{1}*Z*=1 for set 2 which is matched at

*r*=

*R*and

_{3}*Z*=

*R*/

_{1}*R*for set 3 matched at

_{2}*r*=

*R*. Hence, we can choose the proper set of the reduced material parameters as desired.

_{1}## 5. Conclusion

## Acknowledgments

## References and links

1. | A. J. Ward and J. B. Pendry, “Refraction and geometry in Maxwell’s equation,” J. Mod. Opt. |

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. | M. Tsang and D. Psaltis, “Magnifying perfect lens and superlens design by coordinate transformation,” http://www.arxiv.org:physics/0708.0262v1, (2007). |

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

6. | H. Chen and C. T. Chan, “Transformation media that rotate electromagnetic fields,” Appl. Phys. Lett , |

7. | M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, and D. R. Smith, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell’s Equations,” http://www.arxiv.org:physics/0706.2452v1, (2007). |

8. | W. Cai, U. K. Chettiar, A. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photonics |

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

**OCIS Codes**

(160.1190) Materials : Anisotropic optical materials

(260.2110) Physical optics : Electromagnetic optics

(260.2710) Physical optics : Inhomogeneous optical media

**ToC Category:**

Physical Optics

**History**

Original Manuscript: February 4, 2008

Revised Manuscript: March 22, 2008

Manuscript Accepted: April 19, 2008

Published: April 28, 2008

**Citation**

Lan Lin, Wei Wang, Chunlei Du, and Xiangang Luo, "A cone-shaped concentrator with varying performances of concentrating," Opt. Express **16**, 6809-6814 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-10-6809

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### References

- A. J. Ward and J. B. Pendry, "Refraction and geometry in Maxwell's equation," J. Mod. Opt. 43, 773-793 (1996). [CrossRef]
- J. B. Pendry, D. Schurig, and D. R. Smith, "Controlling Electromagnetic Fields," Science 312, 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, 977-980 (2006). [CrossRef] [PubMed]
- M. Tsang and D. Psaltis, "Magnifying perfect lens and superlens design by coordinate transformation," http://www.arxiv.org:physics/0708.0262v1 (2007).
- 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]
- H. Chen and C. T. Chan, "Transformation media that rotate electromagnetic fields," Appl. Phys. Lett. 90, 241105 (2007). [CrossRef]
- M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, and D. R. Smith, " Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell's Equations," http://www.arxiv.org:physics/0706.2452v1. (2007).
- W. Cai, U. K. Chettiar, A. Kildishev, and V. M. Shalaev, "Optical cloaking with metamaterials," Nat. Photonics 1, 224-227 (2007). [CrossRef]
- S. A. Cummer, B. I. Popa, D. Schurig, and D. R. Smith, "Full-wave simulations of electromagnetic cloaking structures," Phy. Rev. E 74, 036621 (2006). [CrossRef]

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