## Analytical study of two-dimensional degenerate metamaterial antennas |

Optics Express, Vol. 19, Issue 15, pp. 13899-13921 (2011)

http://dx.doi.org/10.1364/OE.19.013899

Acrobat PDF (3481 KB)

### Abstract

Dispersion curves of metamaterial steerable antennas composed of two-dimensional arrays of metallic unit structures with the *C*_{4v
} and *C*_{6v
} symmetries are calculated both qualitatively by the tight-binding approximation and quantitatively by the finite-difference time-domain method. Special attention is given to the case of eigenmodes of the *E* symmetry of the *C*_{4v
} point group and those of the *E*_{1} and *E*_{2} symmetries of the *C*_{6v
} point group, since they are doubly degenerate on the Γ point of the Brillouin zone so that they naturally satisfy the steerability condition. We show that their dispersion curves have quadratic dependence on the wave vector in the vicinity of the Γ point. To get a linear dispersion, which is advantageous for steerable antennas, we propose a method of *controlled symmetry reduction*. The present theory is an extension of our previous one [Opt. Express **18**, 27371 (2010)] to two-dimensional systems, for which we can achieve the deterministic degeneracy due to symmetry and the *controlled symmetry reduction* becomes available. This design of metamaterial steerable antennas is advantageous in the optical frequency.

© 2011 OSA

## 1. Introduction

1. V. G. Veselago, “Electrodynamics of substances with simultaneously negative values of sigma and mu,” Sov. Phys. Usp. **10**, 509–514 (1968). [CrossRef]

7. S. A. Ramakrishna and T. M. Grzegorczyk, *Physics and Applications of Negative Refractive Index Materials* (SPIE Press, 2008). [CrossRef]

8. S. Matsuzawa, K. Sato, Y. Inoue, and T. Nomura, “W-band steerable composite right/left-handed leaky wave antenna for automotive applications,” IEICE Trans. Electron. **E89-C**, 1337–1344 (2006). [CrossRef]

*ω*is propagated into a one- or two-dimensional regular array of metallic unit structures that have two dispersion curves in the relevant frequency range. Two important characters for beam steerability are (1) the upper dispersion curve is concave-up and the lower one is concave-down and (2) the two curves touch each other on the Γ point of the Brillouin zone. Then, the incident wave excites an internal eigenmode with a positive group velocity (measured in the direction of incidence). If the frequency of the internal eigenmode is located between

_{i}*ω*and

_{u}*ω*, or above the light lines (see Fig. 1), the incident wave is diffracted at a diffraction angle

_{l}*θ*that is determind by

*ω*and the wave vector of the internal mode

_{i}*k*: which is measured from the direction of incidence. So, when

_{i}*ω*is decreased from

_{i}*ω*to

_{u}*ω*,

_{l}*θ*changes from 0 to 180 degrees. In particular, if

*k*is negative, which takes place when

_{i}*ω*is in the frequency range of the lower dispersion curve, negative diffraction occurs.

_{i}12. K. Sakoda and H.-F. Zhou, “Role of structural electromagnetic resonances in a steerable left-handed antenna,” Opt. Express **18**, 27371–27386 (2010). [CrossRef]

*k*.

*A*

_{1},

*B*

_{1}} or {

*A*

_{2},

*B*

_{2}} symmetries where the two modes in the braces have the same

*σ*parity, dispersion curves are linear in

_{y}*k*and have the slopes of the same magnitude but with different signs in the vicinity of the Γ point. Dispersion curves of this case are illustrated in Fig. 2. In the second case, which is given by combinations of {

*A*

_{1},

*A*

_{2}} or {

*B*

_{1},

*B*

_{2}} symmetries where the two modes in the braces have different

*σ*parities, the dispersion curves are quadratic in

_{y}*k*and have vanishing slopes around the Γ point as was shown previously in Fig. 1.

*θ*varies rapidly with incidence frequency

*ω*, which can be easily understood from a rapid variation of

_{i}*k*with

_{i}*ω*. This rapid variation results in a bad tunability of the diffraction angle, which is an undesirable feature of steerable antennas. On the other hand, if the slopes of the two dispersion curves are finite and have the same magnitude on the Γ point as shown in Fig. 2, variation of

_{i}*θ*with

*ω*is more constant, which results in a better tunability. Therefore, realization of linear dispersion has been one of the main issues of metamaterial steerable antennas.

_{i}*μ*m, so that the antennas have to be fabricated by sophisticated lithographical methods like FIB (focused ion beam) milling and EB (electron beam) lithography. In this case, it may be difficult to adjust the device structure to achieve accidental degeneracy.

*k*dependence is realized. A brief summary is given in Section 5. Massive calculations on mutual relations among electromagnetic transfer integrals,

## 2. Tight-binding calculation

**H**(

**r**,

*t*) is given by where

*ɛ*(

**r**) is the position-dependent dielectric constant of the system and

*c*is the light velocity in free space. The magnetic permeability was assumed to be unity, since we do not deal with magnetic materials. We impose the periodic boundary condition on

**H**to make our problem well-defined [13] and assume, as an approximation to extract basic features of our problem, that

*ɛ*(

**r**) is real. Thus, Eq. (3) leads to eigen value problems.

*ɛ*(

_{s}**r**), we assume a doubly degenerate resonant state due to the geometrical symmetry of the unit structure, and denote its two magnetic field distributions by

*E*symmetry of the

*C*

_{4v}point group and those of the

*E*

_{1}and

*E*

_{2}symmetries of the

*C*

_{6v}point group, as we describe in detail later. Thus, the two distributions satisfy the following eigen value equation: where

*ω*

_{0}is the resonance angular frequency. Without loss of generality, we can assume that the two distributions are normalized as follows: where

*V*is the volume on which we impose the periodic boundary condition. Note that

*i*= 1, 2) is dimensionless by this definition.

*ω*

_{0}, we can assume in the tight-binding picture that the Bloch wave function is a linear combination of

**k**is the wave vector in the first Brillouin zone,

*N*is the number of unit structures in volume

*V*, and

**r**

*is the lattice vector.*

_{nm}*n*and

*m*are integers to denote the two-dimensional lattice points.

*C*

_{4v}and

*C*

_{6v}symmetries in the following, since these two symmetries allow the symmetry-induced degeneracy.

### 2.1. Square lattice

*C*

_{4v}symmetry whose symmetry operations are illustrated in Fig. 3. They consist of identity operation (

*E*), rotation by 90 degrees (

*C*

_{4},

*C*

_{2}), basic mirror reflection (

*σ*,

_{x}*σ*), and diagonal mirror reflection (

_{y}*σ′*,

_{d}*σ″*).

_{d}*C*

_{4v}symmetry, the whole system of the square lattice has the

*C*

_{4v}symmetry. Then, the Bloch states on the Γ point are classified into four one-dimensional representations (

*A*

_{1},

*A*

_{2},

*B*

_{1}, and

*B*

_{2}) and one two-dimensional representation (

*E*). The first four are non-degenerate, while the last is doubly degenerate. Their characters are listed in Table 2. In the rest of this section, we only consider the

*E*mode.

**a**

_{1}and

**a**

_{2}are elementary lattice vectors of the square lattice: Note that

**H**(

_{k}**r**) given by Eq. (6) satisfies the Bloch condition:

**H**(

_{k}**r**) is an eigen function of Eq. (3) and we denote its eigen angular frequency by

*ω*. Thus it satisfies the following equation: where we introduced a differential operator

_{k}*ℒ*: Multiplying Eq. (10) by

*V*, we obtain where

*E*mode can be assumed to transform like the two coordinates,

*x*and

*y*, when any symmetry operation

*R*∈

*C*

_{4v}is applied [14

14. T. Inui, Y. Tanabe, and Y. Onodera, *Group Theory and its Applications in Physics* (Springer, 1990). [CrossRef]

*n*,

*m*) = (0, 0)), the nearest neighbor ((

*n*,

*m*) = (±1, 0), (0, ±1)), and the second nearest neighbor ((

*n*,

*m*) = (±1, ±1)) lattice points in Eq. (13): Because essential features of dispersion curves, especially those around the Γ point, can be described well with a small number of

*n*and

*m*in Eq. (6), we only consider contributions from these lattice points in the following.

*V*, we obtain By solving the secular equation derived from Eqs. (21) and (22), we finally obtain the following dispersion relation:

*k*=

_{x}*k*= 0), we have a doubly degenerate eigen frequency

_{y}*ω*

_{Γ}: Secondly, Eq. (23) is invariant when we exchange

*k*with

_{x}*k*, as it should be, which is a consequence of the

_{y}*C*

_{4v}symmetry of the square lattice. Thirdly, it has a property which is a consequence of the time-reversal symmetry of the wave equation [13].

*k*|, |

_{x}a*k*| ≪ 1 and keep the lowest order terms. Then, from Eq. (23), we obtain where So, the dispersion curves have a quadratic dependence on the wave vector in the vicinity of the Γ point. Their effective mass, which is inversely proportional to the second-order derivative of

_{y}a*ω*

**with respect to the wave vector, is periodic in ϕ with a period of**

_{k}*π*/2.

*k*and

_{x}*k*. So, as far as the behavior of dispersion curves in the vicinity of the Γ point is concerned, they do not bring about a qualitative change.

_{y}### 2.2. Triangular lattice

*C*

_{6v}symmetry whose symmetry operations are illustrated in Fig. 4. They consist of identity operation (

*E*), rotation by 60 degrees (

*C*

_{6},

*C*

_{3}and

*C*

_{2}), and two sets of three equivalent mirror reflections ((

*σ*,

_{x}*σ′*,

_{x}*σ″*) and (

_{x}*σ*,

_{y}*σ′*,

_{y}*σ″*)).

_{y}*C*

_{6v}symmetry, the whole system of the triangular lattice has the

*C*

_{6v}symmetry. Then, the Bloch states on the Γ point are classified into four one-dimensional representations (

*A*

_{1},

*A*

_{2},

*B*

_{1}, and

*B*

_{2}) and two two-dimensional representations (

*E*

_{1}and

*E*

_{2}) [14

14. T. Inui, Y. Tanabe, and Y. Onodera, *Group Theory and its Applications in Physics* (Springer, 1990). [CrossRef]

*E*

_{1}and

*E*

_{2}modes.

*k*, Eq. (42) is approximated as where So, the dispersion curves are isotropic in the vicinity of the Γ point, and they are quadratic in

*k*.

## 3. Numerical results

16. D. M. Sullivan, *Electromagnetic Simulation Using the FDTD Method* (IEEE Press, 2000). [CrossRef]

12. K. Sakoda and H.-F. Zhou, “Role of structural electromagnetic resonances in a steerable left-handed antenna,” Opt. Express **18**, 27371–27386 (2010). [CrossRef]

8. S. Matsuzawa, K. Sato, Y. Inoue, and T. Nomura, “W-band steerable composite right/left-handed leaky wave antenna for automotive applications,” IEICE Trans. Electron. **E89-C**, 1337–1344 (2006). [CrossRef]

*z*component of the magnetic field,

*H*, of resonance states. The resonance states were identified by finding peaks (resonance frequencies) in the Fourier transform of temporal variation of the electromagnetic field after pulsed excitation, whereas their field distributions were obtained by CW (continuous wave) excitation at the resonance frequencies [12

_{z}12. K. Sakoda and H.-F. Zhou, “Role of structural electromagnetic resonances in a steerable left-handed antenna,” Opt. Express **18**, 27371–27386 (2010). [CrossRef]

*E*mode found at 158 GHz for the unit structure of the square symmetry given in Fig. 5(a). These two distributions were obtained by imposing the following boundary conditions on the electromagnetic field according to the symmetry of corresponding polynomial representations of the

*E*mode given in Appendix A: (

*σ*,

_{x}*σ*) = (−1, 1) for (a) and (1, −1) for (b). As is apparent, one distribution is obtained by rotating the other by 90 degrees.

_{y}*E*

_{1}mode found at 177 GHz for the unit structure of the hexagonal symmetry given in Fig. 5(b). In this case, we imposed the same boundary conditions as the

*E*mode according to the symmetry of the polynomial representations of the

*E*

_{1}mode given in Appendix B. Note that one distribution is

*not*obtained by rotating the other for this case. Finally, Figs. 6(e) and 6(f) are the distributions of an

*E*

_{2}mode found at 295 GHz. To obtain these distributions, we imposed the following boundary conditions according to the symmetry of the polynomial representations of the

*E*

_{2}mode given in Appendix C: (

*σ*,

_{x}*σ*) = (−1, −1) for (e) and (1, 1) for (f).

_{y}*E*, which is perpendicular to the surface of the metallic unit structure, because the tangential components (

_{z}*E*and

_{x}*E*) are small due to their continuity across the surface and the small electric field inside the metal caused by the large conductivity. So, the electromagnetic field has mostly a TM (transverse magnetic) character, and therefore,

_{y}*H*is small. Nevertheless, we plotted

_{z}*H*in Fig. 6 in order to provide a graphic description consistent with the analytical calculations given in Section 2.

_{z}*R*by

*χ*

^{(E)}(

*R*) and

*χ*

^{(H)}(

*R*), respectively, then where det

*R*is the determinant of the transformation matrix for

*R*. det

*R*is equal to 1 for proper transformations like rotations and is equal to −1 for improper transformations like mirror reflections. This difference should be treated appropriately when we impose symmetric and antisymmetric boundary conditions on the electromagnetic field to extract resonances and field distributions of particular symmetries.

*E*,

*E*

_{1}, and

*E*

_{2}modes shown in Fig. 6 are considerably larger than those values obtained by our previous analysis of a similar structure [12

**18**, 27371–27386 (2010). [CrossRef]

*E*resonance state shown in Figs. 6(a) and 6(b). They are degenerate on the Γ point as we expected. Their dispersion is consistent with the quadratic dependence on the wave vector in the vicinity of the Γ point, which was predicted by the tight-binding calculation in Section 2.1. In addition to these modes, we also have one mode originating from a non-degenerate

*B*

_{2}resonance state and another mode that has a character of the lowest TM mode of the dielectric slab, whose original dispersion curve is located very close to the light line given by

*ω*=

*ck*.

*electric*field with respect to the

*y*coordinate is denoted by

*p*. We should note that it is opposite to that of the magnetic field. Three of the four modes have the same

_{y}*p*, so that they mix with each other when their dispersion curves come close and show apparent anti-crossing behaviors.

_{y}## 4. Controlled symmetry reduction

*k*-linear terms in the Taylor expansion of

*ω*

**with respect to**

_{k}*k*and

_{x}*k*are relevant to some vanishing

_{y}*C*

_{4v}to

*C*. The latter is a simple point group that consists of identity (

_{s}*E*) and a mirror reflection (

*σ*). Its characters are given in Table 4. It is known by group theory that the

_{x}*E*mode of

*C*

_{4v}splits into an

*A*mode and a

*B*mode of

*C*due to the lowered symmetry.

_{s}*E*mode, and construct a Bloch function with the split

*E*mode functions alone. We denote them by

**H**

_{1}(

*A*mode) and

**H**

_{2}(

*B*mode). They satisfy Eq. (4) as before but with different eigen frequencies: The Bloch function is given by the linear combination of

**H**

_{1}and

**H**

_{2}:

*C*, we have a smaller number of relations among

_{s}*k*=

_{x}*k*= 0 in Eq. (58), the condition for degeneracy is In this case, the dispersion relation is somewhat simplified:

_{y}*k*|, |

_{x}a*k*| ≪ 1 and only keep dominant terms. The result is which has linear dependence on

_{y}a*k*as we expected.

_{x}*C*

_{4v}and

*C*cases. Due to lowered symmetry,

_{s}*Cs*, while they are vanishing for

*C*

_{4v}. These terms, which are represented by

*L*

_{1}in Eqs. (58), (60), and (61), apparently give the

*k*-linear term.

*k*dependence is desirable for application to steerable antennas as we mentioned in Section 1. Compared with the case of purely accidental degeneracy, the two frequencies on the Γ point are close to each other due to their original degeneracy in the case of controlled symmetry reduction. So, we may expect that their frequency tuning can be attained relatively easily.

## 5. Conclusion

*C*

_{4v}symmetry, the

*E*modes are doubly degenerate on the Γ point and their dispersion shows quadratic dependence on the wave vector. In the triangular lattice of the

*C*

_{6v}symmetry, dispersion curves of the

*E*

_{1}and

*E*

_{2}modes, which are also doubly degenerate on the Γ point, are isotropic and have quadratic dependence on the wave vector around the Γ point. To achieve linear

*k*dependence, which is desirable for application to steerable antennas, we proposed a method of

*controlled symmetry reduction*. As an example, we examined the case of symmetry reduction from

*C*

_{4v}to

*C*and derived the dispersion relation of the latter, which showed linear

_{s}*k*dependence as we expected. To verify the analytical calculation, we presented some numerical results of the resonant states of unit structures of square and hexagonal symmetries together with dispersion curves of a square lattice. The theory given in this paper is an extension of our previous one [12

**18**, 27371–27386 (2010). [CrossRef]

*controlled symmetry reduction*becomes available.

## A. *E* mode of *C*_{4v} symmetry

*E*mode is given by a pair of {

*x*,

*y*} [14

14. T. Inui, Y. Tanabe, and Y. Onodera, *Group Theory and its Applications in Physics* (Springer, 1990). [CrossRef]

*E*symmetry,

*R*∈

*C*

_{4v}is operated. By definition, they are transformed as So, for example, We write these two equations in matrix form: In this way, we can introduce the matrix representation of

*R*∈

*C*

_{4v}:

*E*representation in Table 2, which are the trace (sum of the diagonal elements) of the above matrices.

*E*representation, they transform like

*f*

_{1}and

*f*

_{2}. But we should note that there is a difference due to the vector nature of the former. Their transformation is defined as

*ℒ*by Then

**r**to

**r′**=

*σ*

_{x}**r**and evaluate

*σ*does not change the size of volume elements, we have

_{x}*σ*(≡

_{x}*E*) for later use. Since

*ℒ′*is defined as It is an operator in the

**r′**coordinate system equivalent to

*ℒ*in the

**r**coordinate system. We can prove

*ℒ′*=

*ℒ*[13], although we do not use this relation in the following. Substituting the first relation in Eq. (67) and Eq. (68), we obtain So, By similar calculation, we obtain

**r**to

**r′**=

*C*

_{4}

**r**: Note that

**r′**=

*σ*

_{y}**r**, So, we obtain By similar calculation, we can prove

**r**to

**r′**=

*C*

_{4}

**r**,

**r**

_{01}=

**r**

_{10}. Using transformations by

*σ*and

_{x}*n, m*) = (±1, ±1), applying all transformations of

*C*

_{4v}to

*R*∈

*C*

_{4v}and

## B. *E*_{1} mode of *C*_{6v} symmetry

*E*mode of the

*C*

_{4v}symmetry, it is known from group theory that two eigen functions of the

*E*

_{1}mode of the

*C*

_{6v}point group can be assumed to transform like the

*x*and

*y*coordinates when any symmetry operation

*R*∈

*C*

_{6v}is applied [14

*Group Theory and its Applications in Physics* (Springer, 1990). [CrossRef]

*C*

_{6v}. The results are as follows:

*n*= 0. By changing the variable of integration from

**r**to

**r′**=

*C*

_{6}

**r**, we obtain,

*σ*for

_{x}*C*

_{2}for

**r**to

*C*

_{2}

**r**for

*σ*, we obtain Thus, On the other hand, applying

_{x}*σ*,

_{x}*C*

_{2}and

*σ*to

_{y}*σ*,

_{x}*C*

_{2}and

*σ*to

_{y}*M*’s, we can derive three relations as follows. First, by applying

_{i}*R*∈

*C*

_{6v}and

## C. *E*_{2} mode of *C*_{6v} symmetry

*xy*,

*x*

^{2}–

*y*

^{2}} is a polynomial representation of the

*E*

_{2}mode. We can derive the matrix representation of all elements of

*C*

_{6v}:

*C*

_{2}rotation is an idendity matrix. As a consequence, the product of matrices for

*C*

_{6}and

*C*

_{3}is also an identity matrix, so that matrices for

*C*

_{6}and

*C*

_{3}are the same, respectively.

*E*

_{1}mode in Appendix B, we can derive various relations. The results are completely the same as the case of the

*E*

_{1}mode shown in Eqs. (31)–(38).

## D. L n m ( i j ) for *C*_{s} symmetry

_{s}

*n*,

*m*) = (0, 0), (±1, 0), and (0, ±1). Since wave functions

**H**

_{1}and

**H**

_{2}are mutually different due to the symmetry reduction from

*C*

_{4v}to

*C*,

_{s}*ω*

_{1}and

*ω*

_{2}are eigen frequencies for an isolated unit metallic structure described by dielectric constant

*ɛ*(

_{s}**r**). Applying

*σ*to

_{x}*σ*to

_{x}*ℒ*. We denote translation by lattice vector

**r**

*by*

_{nm}*T*. Then, from the translational symmetry of

_{nm}*ɛ*(

**r**) and invariance of differential operators under any uniform translation, we have On the other hand, from the symmetric form of

*ℒ*and the periodic boundary condition, we can prove that

*ℒ*is an Hermitian operator [13]: where

**Q**

_{1}and

**Q**

_{2}are arbitrary vector functions that satisfy the periodic boundary condition.

**r**to

**r′**=

**r**–

**r**

_{10}in the expression of

*M*

_{2}and

*M*

_{3}are real numbers.

*σ*and the same translation, we obtain Similarly for

_{x}*σ*yields identity equations. On the other hand, by changing the variable of integration from

_{x}**r**to

**r**–

**r**

_{01}, we obtain Finally, applying

*σ*, we can prove

_{x}## Acknowledgments

## References and links

1. | V. G. Veselago, “Electrodynamics of substances with simultaneously negative values of sigma and mu,” Sov. Phys. Usp. |

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

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. | R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science |

5. | 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 |

6. | D. R. Smith, S. Schultz, P. Markos, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B |

7. | S. A. Ramakrishna and T. M. Grzegorczyk, |

8. | S. Matsuzawa, K. Sato, Y. Inoue, and T. Nomura, “W-band steerable composite right/left-handed leaky wave antenna for automotive applications,” IEICE Trans. Electron. |

9. | A. Grbic and G. V. Eleftheriades, “Experimental verification of backward-wave radiation from a negative refractive index metamaterial,” J. Appl. Phys. |

10. | C. Caloz and T. Ito, “Application of the transmission line theory of left-handed (LH) materials to the realization of a microstrip LH line,” IEEE-AP-S Int. Symp. Dig. |

11. | S. Tokoro, K. Kuroda, A. Kawakubo, K. Fujita, and H. Fujinami, “Electronically scanned millimeter-wave radar for pre-crush safety and adaptive cruise control system,” Proc. IEEE Intelligent Vehicles Symp. , 304–309 (2003). |

12. | K. Sakoda and H.-F. Zhou, “Role of structural electromagnetic resonances in a steerable left-handed antenna,” Opt. Express |

13. | K. Sakoda, |

14. | T. Inui, Y. Tanabe, and Y. Onodera, |

15. | A. Taflove, |

16. | D. M. Sullivan, |

**OCIS Codes**

(280.5600) Remote sensing and sensors : Radar

(350.4010) Other areas of optics : Microwaves

(350.3618) Other areas of optics : Left-handed materials

(160.3918) Materials : Metamaterials

(160.5298) Materials : Photonic crystals

**ToC Category:**

Metamaterials

**History**

Original Manuscript: April 26, 2011

Revised Manuscript: June 16, 2011

Manuscript Accepted: June 17, 2011

Published: July 6, 2011

**Citation**

Kazuaki Sakoda and Haifeng Zhou, "Analytical study of two-dimensional degenerate metamaterial antennas," Opt. Express **19**, 13899-13921 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-15-13899

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### References

- V. G. Veselago, “Electrodynamics of substances with simultaneously negative values of sigma and mu,” Sov. Phys. Usp. 10, 509–514 (1968). [CrossRef]
- J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006). [CrossRef] [PubMed]
- 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, 4184–4187 (2000). [CrossRef] [PubMed]
- R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001). [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]
- D. R. Smith, S. Schultz, P. Markos, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65, 195104 (2002).
- S. A. Ramakrishna and T. M. Grzegorczyk, Physics and Applications of Negative Refractive Index Materials (SPIE Press, 2008). [CrossRef]
- S. Matsuzawa, K. Sato, Y. Inoue, and T. Nomura, “W-band steerable composite right/left-handed leaky wave antenna for automotive applications,” IEICE Trans. Electron. E89-C, 1337–1344 (2006). [CrossRef]
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