## Role of structural electromagnetic resonances in a steerable left-handed antenna |

Optics Express, Vol. 18, Issue 26, pp. 27371-27386 (2010)

http://dx.doi.org/10.1364/OE.18.027371

Acrobat PDF (4378 KB)

### Abstract

We reformulate the problem of a steerable left-handed antenna reported by Matsuzawa et al. [IEICE Trans. Electron. **E89-C**, 1337 (2006)] from the view point of structural electromagnetic resonance of the unit structure. We show that there are two such resonances with different spatial symmetries in the relevant frequency range, which result in the formation of two electromagnetic bands with opposite signs of curvature at the Γ point of the Brillouin zone. We derive an expression of dispersion curves based on the tight-binding picture and show that the dispersion of the two bands is linear in the vicinity of the Γ point in the case of accidental degeneracy only if the symmetry of the two resonance states satisfies certain conditions. We also show that the refraction angle can be designed by changing the lattice constant of the arrayed unit structures, since the band width is modified due to the change in the electromagnetic *transfer integral*.

© 2010 Optical Society of America

## 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]

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]

## 2. Theory

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]

*C*

_{2v}symmetry. Since the antenna was designed to operate at 76 GHz (free-space wavelength = 3.95 mm), the size of the unit structure is of the order of 1 mm and the period of the array, or the lattice constant

*a*, is 0.6 mm. The antenna structure is periodic in the

*x*direction. An incident wave with angular frequency

*ω*is supplied in the positive

_{i}*x*direction to propagate in the dielectric slab.

*x*coordinate. The presence of such resonant states will be proved in Section 3.

*B*in the one-dimensional tight-binding model in the lowest order approximation is given as follows: where

*ϕ*(

**r**) is a normalized atomic wave function on the lattice point at the origin,

*ϕ*(

**r**–

**a**) is that on the nearest neighbor lattice point

**a**, and

*U*(

**r**) (< 0) is the attractive potential from all atomic nuclei (ionic cores) except that at the origin [17]. For simplicity, we ignored overlapping of wave functions at the second nearest neighbor and farther lattice points in Eq. (1). The energy band is given by where

*E*

_{0}is the sum of the original electronic energy of an isolated atom and If the wave function is of the s character, that is, if its parity is even, then

*ϕ*(

**r**) and

*ϕ*(

**r**–

**a**) have the same sign at

**r**=

**a**/2 where the dominant contribution is made to the integral of Eq. (1), so that

*B*> 0. On the other hand, if the wave function is of the p character and has the odd parity,

*B*< 0.

*f*(

_{l}*f*),

_{u}*A*(

_{l}*A*), and

_{u}*ν*(

_{l}*ν*) are the eigen angular frequency, band width, and central angular frequency of the lower (upper) band, respectively. Here we should note that the square of the eigen frequency takes the role of the electronic energy, since Maxwell’s wave equation is of the second order for the time variable whereas the Schrödinger equation is of the first order. We further assume so that the two bands have the same frequency at the Γ point (

_{u}*k*= 0). This is the second condition that the steerable antenna should satisfy. As will be shown in Section 3, this condition can be realized, for example, by adjusting the lattice constant.

*k*space, which results in the linear dispersion in the vicinity of the Γ point. This case will be described in the next section.

*ɛ*(

**r**), for which we assume the Drude function in the metallic region, where

*ω*is the angular frequency and

*ω*is the plasma frequency. We assume that magnetic permeability is equal to unity, since we do not deal with magnetic materials. Then, from the Maxwell equations, we have the wave equation for the magnetic field

_{p}**H**(

**r**

*,t*): where

*c*is the light velocity in free space. For a single unit structure, we denote its dielectric constant by

*ɛ*(

_{s}**r**) and assume that we know its eigen function

**H**

_{0}(

**r**) with eigen frequency

*ω*

_{0}and that it is normalized: where

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

**H**

_{0}is dimensionless by this definition. Then we approximate the eigen function of the periodic system

**H**

*(*

_{k}**r**) by a linear combination of

**H**

_{0}(

**r**–

*n*

**a**) (

*n*= 1, 2, ··· ,

*N*): where

**a**is (

*a*, 0, 0). In the lowest order approximation, eigen frequency of the Bloch band

*ω*is given by where

_{k}*δχ*(

**r**) is the difference of 1/

*ɛ*(

**r**) between the periodic system and a single unit structure: which is negative or equal to zero for

*ω*<

*ω*. Performing integration by parts, the electromagnetic

_{p}*transfer integral L*

_{1}is given by

*ɛ*

_{0}is the permittivity of free space. In Eq. (16), we introduced the eigen function of the electric field

**E**

_{0}given by The first term on the right-hand side of Eq. (16) is the integral of the normal component of the integrand on the surface

*S*of volume

*V*, and is vanishing due to the periodic boundary condition. Since a localized mode is a standing wave, its phase is the same everywhere. Thus we can choose

**E**

_{0}(

**r**) as a real function. Because

*δχ*(

**r**) is non-zero only in the region where

*ɛ*(

_{s}**r**) = 1 (air), we finally obtain

**E**

_{0}(

**r**) wave function is expected to keep its sign in the region between two adjacent unit structures. Thus we can follow a similar logic as in the case of p and s electronic states in semiconductor. However, we should take into consideration two differences. The first one is the different transformation properties between electronic wave functions and electromagnetic wave functions, or between scalar waves and vector waves. When we denote the mirror reflection of the

*x*coordinate about the origin by

*σ*, then the symmetry of an electronic wave function

_{x}*ϕ*(

**r**) is expressed as [18] where the + sign and – sign correspond to even and odd parities, respectively. On the other hand, the symmetry of an electric field

**E**(

**r**) is given by [18] So, for the even parity case, For the odd parity, we have opposite signs on the right-hand side of these three equations. The second difference is that because of the continuity of the normal component of the electric displacement and the negative dielectric constant of the metallic unit structure with the Drude dielectric function, an additional minus sign has to be taken into consideration when we relate the inner normal component of the electric field with the outer one across the metal surface.

*E*is dominant on the metal surface and so is it at the nearest neighbor lattice points. Thus, Eq. (18) can be approximated by We can conclude that if

_{z}**E**

_{0}has even (odd) parity with respect to the

*x*coordinate,

*L*

_{1}is positive (negative) and the Bloch band is concave-down (up).

*ω*= ±

*ck*are also plotted. Note that the horizontal axis of this figure is the normalized crystalline momentum (divided by

*h̄*) in the

*x*direction, and is

*not*the wave vector in free space.

*ω*is propagated in the dielectric slab in the positive

_{i}*x*direction, an internal mode with crystalline momentum

*k*is excited, and a diffracted wave is generated. Here we should note that an internal mode with a positive group velocity (=

_{i}*df*/

_{l,u}*dk*), which is equal to the energy velocity [19

19. P. Yeh, “Electromagnetic propagation in birefringent layered media,” J. Opt. Soc. Am. **69**, 742–756 (1979). [CrossRef]

*k*’s are present for each

_{i}*ω*, only the positive (negative)

_{i}*k*mode of the upper (lower) band can be excited by the incident wave.

_{i}*z*component of the diffracted wave by

*k*, since

_{z}*k*should be a real number (See Fig. 3). So, diffraction is possible only in the region above the light line as for photonic crystal slabs [13], whose lower and upper bounds are labelled

_{z}*ω*and

_{l}*ω*in Fig. 2. Then, the diffraction angle

_{u}*θ*is given by for the lower band and for the upper band, where the value range of cos

^{−1}is from 0 to

*π*.

*k*is limited by two light lines (see Fig. 2), the emission angle always spans from 0 to 180 degrees.

**E89-C**, 1337–1344 (2006). [CrossRef]

*ω*was fixed. On the other hand, an additional dielectric plate was placed above the metallic array, and the distance between the plate and the array was varied so that the mode frequencies (dispersion curves) were shifted continuously by the perturbation induced by the dielectric plate. Thus the relative position of the incident frequency to the dispersion curves was adjusted and the diffraction angle changed accordingly.

_{i}^{++}codes with the conventional Yee mesh and the convolutional PML (perfectly matched layer) boundary conditions. Since we were dealing with the microwave range, we assumed a low frequency limit of the analytical Drude function for metallic

*ɛ*: where

*δ*is a positive infinitesimal to assure the causality,

*γ*is the damping constant, and

^{7}S/m for

*σ*(value for copper).

**E**and

**H**, and calculate

**H**

_{(}

_{n}_{+1/2)}

_{Δ}

*and*

_{t}**E**

_{n}_{Δ}

*, where*

_{t}*n*is an integer and Δ

*t*is the time-discretization unit. When we apply this procedure to Ampere’s law, we discretize the right-hand side such that to avoid the convolution formula for frequency dependent dielectric constants [15]. The metallic unit structure was assumed to be sufficiently thin and treated as one layer of Yee’s mesh.

*k*-dependent resonance frequencies by searching peaks in the Fourier transform of the temporal variation of the electromagnetic field after pulsed excitation.

## 3. Results and discussion

*x*coordinate, whereas the lower-frequency resonance has an even parity. As a result, we may assume that the higher band is concave-up and the lower band is concave-down according to the discussion in the previous section, although the assumption of negative

*ɛ*does not rigorously apply to the present case. Nonetheless, the above assumption actually happened as shown in Fig. 6, where among the two dispersion curves, the upper one is concave-up and the lower one is concave-down at the Γ point.

*C*

_{2}

*symmetry, eigenmodes at the Γ point are all one-dimensional, i.e., non-degenerate [18]. So, the coincidence of eigen frequencies of the two modes, if any, is accidental degeneracy, which is brought about by a particular choice of sample parameters. The case of accidental degeneracy will be described later. Thus the probability of having different frequencies is 100 % from the purely mathematical point of view. However, as will also be shown later, we can make them as close to each other as we like by adjusting sample parameters.*

_{v}*y*coordinate and showed the anti-crossing behavior that occurred due to intermixing when its frequency crossed the dispersion curves of the two flat bands. The third band originated from the lowest TM (transverse magnetic) mode of the dielectric slab whose dispersion is given by the solution of the following secular equation. where

*ɛ*and

_{d}*d*are the dielectric constant and thickness of the slab waveguide, and

*k*and

_{z}*κ*are given by As is also apparent from this anti-crossing behavior, the band modes shown in Fig. 6 can be excited by incident waves of the TM polarization.

*C*

_{2}

*point group. By consulting its character table given in Table 1, it is found that the lower- and higher-frequency modes belong to the*

_{v}*A*

_{1}and

*B*

_{1}representations, respectively. As for the symmetry of the band states, it is the same as the original resonant states on the Γ point (

*k*= 0) and the boundary of the Brillouin zone (

*k*= ±

*π*/

*a*). On the other hand, it only keeps the parity with respect to the

*y*coordinate in the rest of the Brillouin zone [18].

*A*

_{1}and

*B*

_{1}modes. Thus the symmetries of the two bands on the Γ point are also

*A*

_{1}and

*B*

_{1}so that they are mutually different and do not repel each other even when they come close by choosing device parameters. On the other hand, since the two bands have the same even parity with respect to the

*y*coordinate for general

*k*, they repel each other when they come close. According to the valance of these two factors, linear dispersion curves are realized in the vicinity of the Γ point in the case of accidental degeneracy.

*ω*

_{1}and

*ω*

_{2}: We also assume that the wave functions are normalized: where

*V*is the volume on which we imposed the periodic boundary condition. Then the Bloch wave function in the tight-binding picture must be a linear combination of

*ɛ*on the left-hand side of Eq. (42). Multiplying

_{s}*V*, we obtain where we kept the lowest order term and ignored those contributions from wave functions on the nearest neighbor and farther lattice points. As for term proportional to

*δχ*, where

*B*that

*k*= 0 in Eq. (51) and assuming the two eigen frequencies are the same, we obtain the following condition for the degeneracy. where the degenerate eigen frequency is denoted by

*ω*

_{Γ}. Then, the secular equation reduces to To examine the dispersion in the vicinity of the Γ point, we assume

*k*≪

*a*and keep the lowest order terms with respect to

*k*. Thus we obtain So, the two dispersion curves are linear in

*k*and have finite slopes whose magnitudes are the same and signs are opposite. In particular, the time reversal symmetry described by Eq. (33) holds. These results coincide with previous findings by the theory of transmission lines [10, 20

20. C. Caloz, A. Lai, and T. Itoh, “The challenge of homogenization in metamaterials,” N. J. Phys. **7**, 167 (2005). [CrossRef]

21. A. Lai, T. Itoh, and C. Caloz, “Composite right/left-handed transmission line metamaterials,” IEEE Microwave Magazine, September issue, 34–50 (2004). [CrossRef]

*A*

_{1}and

*B*

_{1}. The same behavior is expected for a combination of

*A*

_{2}and

*B*

_{2}states, since they have the same odd parity with respect to the

*y*coordinate. The remaining two combinations, (

*A*

_{1},

*B*

_{2}) and (

*A*

_{2},

*B*

_{1}), have different

*σ*parity so that the corresponding bands do not mix and they merely cross each other.

_{y}*L*

_{1}as Eqs. (12) and (14) show. When the lattice constant is increased, the overlapping of wave functions of the nearest neighbors decreases so that the band width also decreases. To confirm this, we calculated the mode frequencies on the Γ point for various lattice constants. The results are plotted in Fig. 8, which clearly shows that (1) the frequency of the

*A*

_{1}(

*B*

_{1}) mode decreases (increases) with increasing lattice constant, which is consistent with the decreasing band width, since the

*A*

_{1}(

*B*

_{1}) band is concave-down (up) on the Γ point, and (2) the mode frequencies approach the resonance frequencies of a single unit structure (82.8 GHz and 96.6 GHz), when the lattice constant becomes sufficiently large so that the influence of neighboring unit structures is vanishing. When the lattice constant is approximately equal to 0.576 mm, the two curves in Fig. 8 cross each other, which implies that the two mode frequencies coincide as shown in Fig. 7(b). When the lattice constant is smaller than this value and the order of mode frequencies is reversed, dispersion curves should look like Fig. 7(c) and (d), since the parities of the two modes with respect to the

*y*coordinate are the same so that the two modes satisfy all conditions to realize the situation of Fig. 7.

## 4. Conclusion

## 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,” |

12. | J. D. Joannopoulos, R. D. Meade, and J. N. Winn. |

13. | K. Sakoda, |

14. | T. Ito and K. Sakoda, “Photonic bands of metallic systems. II. Features of surface plasmon polaritons,” Phys. Rev. B |

15. | A. Taflove, |

16. | D. M. Sullivan, |

17. | N. Peyghambarian, S. W. Koch, and A. Mysyrowicz, |

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

19. | P. Yeh, “Electromagnetic propagation in birefringent layered media,” J. Opt. Soc. Am. |

20. | C. Caloz, A. Lai, and T. Itoh, “The challenge of homogenization in metamaterials,” N. J. Phys. |

21. | A. Lai, T. Itoh, and C. Caloz, “Composite right/left-handed transmission line metamaterials,” IEEE Microwave Magazine, September issue, 34–50 (2004). [CrossRef] |

**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: November 4, 2010

Revised Manuscript: December 8, 2010

Manuscript Accepted: December 9, 2010

Published: December 13, 2010

**Citation**

Kazuaki Sakoda and Haifeng Zhou, "Role of structural electromagnetic resonances in a steerable left-handed antenna," Opt. Express **18**, 27371-27386 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-26-27371

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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). [CrossRef]
- 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. E 89-C, 1337-1344 (2006). [CrossRef]
- A. Grbic, and G. V. Eleftheriades, "Experimental verification of backward-wave radiation from a negative refractive index metamaterial," J. Appl. Phys. 92, 5930-5935 (2002). [CrossRef]
- 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. 2, 412-415 (2002).
- 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).
- J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University Press, Princeton, 1995).
- K. Sakoda, Optical Properties of Photonic Crystals, 2nd Ed. (Springer-Verlag, Berlin, 2004).
- T. Ito, and K. Sakoda, "Photonic bands of metallic systems. II. Features of surface plasmon polaritons," Phys. Rev. B 64, 045117 (2001). [CrossRef]
- A. Taflove, Computational Electrodynamics (Artech House, Boston, 1995).
- D. M. Sullivan, Electromagnetic Simulation Using the FDTD Method (IEEE Press, Piscataway, 2000). [CrossRef]
- N. Peyghambarian, S. W. Koch, and A. Mysyrowicz, Introduction to Semiconductor Optics (Prentice Hall, Englewood Cliffs, 1993) Sec. 2.5.
- T. Inui, Y. Tanabe, and Y. Onodera, Group Theory and Its Applications in Physics (Springer, Berlin, 1990).
- P. Yeh, "Electromagnetic propagation in birefringent layered media," J. Opt. Soc. Am. 69, 742-756 (1979). [CrossRef]
- C. Caloz, A. Lai, and T. Itoh, "The challenge of homogenization in metamaterials," N. J. Phys. 7, 167 (2005). [CrossRef]
- A. Lai, T. Itoh, and C. Caloz, "Composite right/left-handed transmission line metamaterials," IEEE Microwave Magazine, September issue, 34-50 (2004). [CrossRef]

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