## Hybrid resonant phenomena in a SRR/YIG metamaterial structure

Optics Express, Vol. 17, Issue 4, pp. 2122-2131 (2009)

http://dx.doi.org/10.1364/OE.17.002122

Acrobat PDF (520 KB)

### Abstract

We consider the hybridization of the resonance of a SRR metamaterial with the gyromagnetic material resonance of yittrium iron garnet (YIG) inclusions. The combination of an artificial structural resonance and natural material resonance generates a unique hybrid resonance that can be harnessed to make tunable metamaterials and further extend the range of achievable electromagnetic materials. A predictive analytic model is applied that accurately describes the characteristics of this SRR/YIG hybridization. We suggest that this hybridization has been observed in experimental data presented by Kang et al. [Opt. Express, **16**, 8825 (2008)] and present numerical simulations to support this assertion. In addition, we investigate a design for optimizing the SRR/YIG structure that shows strong hybridization with a minimum amount of YIG material.

© 2009 Optical Society of America

## 1. Introduction

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

3. N. Seddon and T. Bearpark, “Observation of the Inverse Doppler Effect,” Science **302**, 1537 (2003). [CrossRef] [PubMed]

4. V. G. Veselago, “The electrodynamics of substances with simultaneously negative *ε* and *μ*,” Soviet Physics Uspekhi **10**, 509–514 (1964). [CrossRef]

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 **314**, 977–980 (2006). http://www.sciencemag.org/cgi/reprint/314/5801/977.pdf. [CrossRef] [PubMed]

6. I. Gil, J. G. Garcia, J. Bonache, F. Martin, M. Sorolla, and R. Marques, “Varactor-loaded split ring resonators for tunable notch filters at microwave frequencies,” Electron. Lett. **40**, 1347–1348 (2004). [CrossRef]

7. S. Lim, C. Caloz, and T. Itoh, “Metamaterial-based electronically controlled transmission-line structure as a novel leaky-wave antenna with tunable radiation angle and beamwidth,” IEEE Trans. Microwave Theory Tech. **52**, 2678–2690 (2004). [CrossRef]

8. T. Hand and S. Cummer, “Frequency Tunable Electromagnetic Metamaterial Using Ferroelectric Loaded Split Rings,” J. Appl. Phys. **103**, 066105 (2007). [CrossRef]

9. H.-T. Chen, W. J. Padilla, J. M. O. Zide, A. C. Gossard, A. J. Tayor, and R. D. Averitt, “Active terahertz metama-terial devices,” Nature **444**, 597–600 (2006). [CrossRef] [PubMed]

10. Q. Zhao, L. Kang, B. Du, B. Li, and J. Zhou, “Electrically tunable negative permeability metamaterials based on nematic liquid crystals,” Appl. Phys. Lett. **90**, 011112 (2007). [CrossRef]

11. V. B. Bregar, “Effective-medium approach to the magnetic susceptibility of composites with ferromagnetic inclusions,” Phys. Rev. B **71**, 174418 (2005). [CrossRef]

12. Y. He, P. He, S. D. Yoon, P. V. Parimi, F. J. Rachford, V. G. Harris, and C. Vittoria, “Tunable negative index metamaterial using yttrium iron garnet,” J. Magn. Magn. Mater. **313**, 187–191 (2007). [CrossRef]

1. L. Kang, Q. Zhao, H. Zhao, and J. Zhou, “Magnetically tunable negative permeability metamaterial composed by split ring resonantors and ferrite rods,” Opt. Express **16**, 8825–8834 (2008). [CrossRef] [PubMed]

1. L. Kang, Q. Zhao, H. Zhao, and J. Zhou, “Magnetically tunable negative permeability metamaterial composed by split ring resonantors and ferrite rods,” Opt. Express **16**, 8825–8834 (2008). [CrossRef] [PubMed]

1. L. Kang, Q. Zhao, H. Zhao, and J. Zhou, “Magnetically tunable negative permeability metamaterial composed by split ring resonantors and ferrite rods,” Opt. Express **16**, 8825–8834 (2008). [CrossRef] [PubMed]

## 2. Theory

14. J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech. **47**, 2075–2084 (1999). [CrossRef]

15. R. Liu, T. J. Cui, D. Huang, B. Zhao, and D. R. Smith, “Description and explanation of electromagnetic behavior in artificial metamaterials based on effective medium theory,” Phys. Rev. E **76**, 026606 (2007). [CrossRef]

*k*(

*ω*) =

*ω*/

*c*is the free-space wavevector and

*d*is the unit cell size of the metamaterial.

*θ*(

*ω*) corresponds to the phase advance per unit cell. The effective wave impedance is then defined,

*ε*

_{eff}, and permeability,

*μ*

_{eff}, are then determined in the usual way,

*μ*

_{eff}=

*n*(

*ω*)

*η*(

*ω*) and

*ε*

_{eff}=

*n*(

*ω*)/

*η*(

*ω*). Given the Drude-Lorentz parameters (

*F*

_{(j)},

*A*

_{(j)}, Γ

_{SRR(j)},

*ω*

_{0SRR(j)}and the unit cell size (

*d*), we can use Eqs. (1–6) to accurately predict the effective electromagnetic response over a wide frequency range.

*μ*

_{1}and

*μ*

_{2}are resonant functions of frequency given by

*γ*the gyromagnetic ratio and

*μ*

_{0}

*M*the saturation magnetization. The resonant frequency is given by

_{s}*ω*

_{0YIG}= (

*γμ*

_{0}

*H*

_{0}- iωΓ

*) where*

_{YIG}*H*

_{0}is the DC magnetic bias field. The loss component, Γ

*, can be further written in terms of the linewidth of the resonance, Δ*

_{YIG}*H*, as Γ

*=*

_{YIG}*γμ*

_{0}Δ

*H*/ (2

*ω*). For our analysis we ignore the off-diagonal components in Eq. (7) which only have a strong influence at resonance. We show that this is a suitable approximation if we restrict our unit cells to include thin YIG sheets.

*x*component of the YIG permeability tensor or

*μ*

_{1}(

*ω*) in Eq. (7). The resulting inductance of the SRR structure can be calculated by abstractly conceptualized a ‘magnetic capacitor’ with parallel current sheets sandwiching a volume containing some fraction of which is filled with YIG material. Considering this simple model, the inductance of the SRR structure is found to be [13],

*q*is a constant that depends on the ‘effective’ volume fraction of the YIG material (

*q*must be determined through a fitting procedure),

*μ*

_{1}(

*ω*) is the relative permeability of the magnetic material, and

*g*

_{geom}is a constant with units of length that is determined by the geometry of the SRR structure. Alternatively, the capacitance of the SRR structure is constant and can be written as,

*C*=

*ε*

_{0}

*h*

_{geom}, where

*h*

_{geom}is a constant with units of length that is determined by the geometry of the structure. If we plug the associated inductance and capacitance of the SRR structure into Eq. (3) we get an equation governing the new

*hybrid*resonant frequencies of the SRR structure,

*ω*

_{0}

*=*

_{SRR}*μ*

_{0}

*ε*

_{0}

*g*

_{geom}

*h*

_{geom}= 2

*π*

*f*

_{0}is the resonant frequency of the SRR structure in the absence of the YIG material. One can replace

*ω*→

*ω*′

*in Eq. (11) to obtain a transcendental equation that can be solved for the new hybrid resonant frequencies [13].*

_{0SRR}*μ*

_{eff}, and permittivity,

*ε*

_{eff}, for the hybrid structure from the modified Drude-Lorentz model by replacing

*ω*

_{0SRR}inEq. (1)and Eq.(2) with Eq. (11). This involves a two-step process of first performing numerical simulations of the SRR structure without the magnetic response of the YIG material, and using a least square fitting procedure of Eqs. (1–6) to determining the Drude-Lorentz parameters. Once the Drude-Lorentz parameters are determined the magnetic filling fraction constant,

*q*, can be determined through comparison of the analytic theory to the numerical simulations of the combined SRR/YIG structure at several magnetic biasing values. Once

*q*is known, the analytic theory provides a predictive model that is valid over the full frequency range of interaction. Further, the fres-nel equations can be solved to determine the analytic transmission and reflection of SRR/YIG metamaterial structure [16

16. 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]

## 3. Numerical Simulations

*μ*

_{0}

*M*= 1.7 kGs, linewidth Δ

_{s}*H*=12 Oe, and permittivity

*ε*= 14.7. A standard full wave analysis of a single unit cell of the SRR/YIG structure, with appropriate boundary conditions, is sufficient to fully characterize the bulk SRR/YIG metamaterial structure’s transmission and reflection coefficients.

16. 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]

*F*

_{(1)}= 0.228,

*ω*

_{0SRR}

_{(1)}= 2

*π*(10.6) Ghz, Γ

_{SRR(1)}= 2

*π*(0.00219) Ghz,

*A*

_{(1)}= 1.13] and the second order resonance [

*F*

_{(2)}= 0.659,

*ω*

_{0SRR(2)}= 2

*π*(23.7) Ghz, Γ

_{SRR(2)}= 2

*π*(0.0347) Ghz,

*A*

_{(2)}= 2.99] were determined by a least square fit using Eqs. (1–6). The inclusion of Eq. (11) was then used in Eq. (1) and Eq. (2) to determine the effective hybridized permeability and permittivity of the structure as a function of the magnetic bias,

*H*

_{0}, and magnetic filling fraction,

*q*. The Fresnel equations were then used to determine the theoretical transmission of the SRR/YIG metama-terial structure. Through a fitting process of the analytic to the simulated transmission, the magnetic filling fraction was determine to be

*q*= 0.06.The comparison of the HFSS calculated transmission and the analytic solution are shown in Fig. 2 for a bias field of

*H*

_{0}= 3 kG which results in a YIG resonance that is near the natural SRR resonance. We see that there is good correlation between analytic solution and simulation in both the position of the resonant frequency and magnitude. The splitting of the fundamental mode is seen, 1

*and 1*

_{a}*, and a small contribution from the splitting of the higher order resonance, 2*

_{b}*, is also seen. The 1*

_{a}*lower resonance deviates somewhat from the magnitude of the analytic theory and this may be the result of the off-diagonal components of the YIG permeability tensor having an effect near resonance. Also plotted is the simulation of the YIG strips by themselves (without the SRR structure) and it has an almost negligible effect on transmission. This confirms that the magnetic properties of the YIG material strongly interact with the local fields generated by the SRR structure in order to create the hybridization.*

_{b}*, of the SRR shifts to the right and a small resonance, 1*

_{b}*, appears and grows to the left. Once the YIG resonance passes the natural SRR resonance, the 1*

_{a}_{b}resonance continues to decline and eventually disappears while the 1

*resonance reclaims the position of the natural SRR resonance. When the YIG and SRR resonance overlap, the hybridization is greatest with two relatively equal resonances straddling the original SRR resonance position. Also shown in Fig. 3 are the extracted permeability and permittivity from the numerical simulations. We further plot the extracted permeability of the hybrid structure over the biasing range*

_{a}*H*

_{0}= 1.7 – 4.5 kG on the same plot in Fig. 2(b). For metamaterial design we see that there is a wide range of tunability that can be harnessed. We can also note the slight asymmetry in the resonant peaks and this may be the result of the off diagonal components in the permeability tensor of the YIG.

**16**, 8825–8834 (2008). [CrossRef] [PubMed]

*F*

_{(1)}= 0.0483,

*ω*

_{0SRR(1)}= 2

*π*(9.97) Ghz, Γ

_{SRR(1)}= 0.108 Ghz,

*C*

_{(1)}= 1.22] and the second order resonance [

*F*

_{(2)}= 0.475,

*ω*

_{0SRR(2)}= 2

*π*(16.2) Ghz, Γ

_{SRR(2)}= 0.0461 Ghz,

*C*

_{(2)}= 7.18] by a least square fit of Eqs. (1–6). As noted above, the double SRR structure used by Kang has a second order resonances that is very close to the fundamental resonance. Also, the large size of the unit cell size, 5 mm, with respect to the SRR structure, 2.2 mm, results in strong spatial dispersion effects. In addition, the significant thickness of the YIG rod used, 0.8 mm, suggest that the off-diagonal components of the permeability tensor of the YIG material will have a significant effect. Further, the significant volume the material takes up in the unit cell suggest that it will have strong direct interaction with incident radiation. Still, the theory developed previously loosely characterizes the resulting hybridization.

*H*0 = 3 – 3.8 kG), possibly due to the off-diagonal permeability components having a strong influence in this structure. We can also observe that the 2

*resonance, which is due to the hybridization of secondary SRR resonance, is much stronger due to the secondary SRR resonance being closer to the fundamental resonance. In addition, the 2*

_{a}*resonance is mixed together with the direct YIG response.*

_{a}*q*= 0.08, the analytic theory roughly correlates with the three major hybrid resonances, though there is a slight offset at some frequencies. We suggest that the messy resonance noted by Kang et al. when the SRR and YIG resonances coincided was in fact the first experimental observation of the hybridization effect.

*q*= 0.06), that is shown Fig. 1, is approximately the same as Kang et al.’s structure, though it utilize 96% less YIG material. This shows the effectiveness of targeting the YIG material in the regions of strong local fields of the SRR structure, i.e. near the wire. The design of the optimized hybrid structure we considered in Fig. 1 could be simplified by considering a single sheet of YIG material extending across the entire unit cell instead of the double stripped structure. We consider the double stripped structure solely to demonstrate that targeting the material in the regions where the local fields of the SRR are strongest maximizes the interaction for the amount of YIG material used.

## 4. Conclusion

## References and links

1. | L. Kang, Q. Zhao, H. Zhao, and J. Zhou, “Magnetically tunable negative permeability metamaterial composed by split ring resonantors and ferrite rods,” Opt. Express |

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

3. | N. Seddon and T. Bearpark, “Observation of the Inverse Doppler Effect,” Science |

4. | V. G. Veselago, “The electrodynamics of substances with simultaneously negative |

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. | I. Gil, J. G. Garcia, J. Bonache, F. Martin, M. Sorolla, and R. Marques, “Varactor-loaded split ring resonators for tunable notch filters at microwave frequencies,” Electron. Lett. |

7. | S. Lim, C. Caloz, and T. Itoh, “Metamaterial-based electronically controlled transmission-line structure as a novel leaky-wave antenna with tunable radiation angle and beamwidth,” IEEE Trans. Microwave Theory Tech. |

8. | T. Hand and S. Cummer, “Frequency Tunable Electromagnetic Metamaterial Using Ferroelectric Loaded Split Rings,” J. Appl. Phys. |

9. | H.-T. Chen, W. J. Padilla, J. M. O. Zide, A. C. Gossard, A. J. Tayor, and R. D. Averitt, “Active terahertz metama-terial devices,” Nature |

10. | Q. Zhao, L. Kang, B. Du, B. Li, and J. Zhou, “Electrically tunable negative permeability metamaterials based on nematic liquid crystals,” Appl. Phys. Lett. |

11. | V. B. Bregar, “Effective-medium approach to the magnetic susceptibility of composites with ferromagnetic inclusions,” Phys. Rev. B |

12. | Y. He, P. He, S. D. Yoon, P. V. Parimi, F. J. Rachford, V. G. Harris, and C. Vittoria, “Tunable negative index metamaterial using yttrium iron garnet,” J. Magn. Magn. Mater. |

13. | J. N. Gollub, D. R. Smith, and J. D. Baena, “Hybrid resonant phenomenon in a metamaterial structure with integrated resonant magnetic material,” arXiv:0810.4871 (2008). |

14. | J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech. |

15. | R. Liu, T. J. Cui, D. Huang, B. Zhao, and D. R. Smith, “Description and explanation of electromagnetic behavior in artificial metamaterials based on effective medium theory,” Phys. Rev. E |

16. | 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 |

**OCIS Codes**

(160.3820) Materials : Magneto-optical materials

(160.3918) Materials : Metamaterials

**ToC Category:**

Metamaterials

**History**

Original Manuscript: October 28, 2008

Revised Manuscript: November 13, 2008

Manuscript Accepted: November 15, 2008

Published: February 2, 2009

**Citation**

Jonah N. Gollub, Jessie Y. Chin, Tie J. Cui, and David R. Smith, "Hybrid resonant phenomena in a SRR/YIG metamaterial structure," Opt. Express **17**, 2122-2131 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-4-2122

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### References

- L. Kang, Q. Zhao, H. Zhao, and J. Zhou, "Magnetically tunable negative permeability metamaterial composed by split ring resonantors and ferrite rods," Opt. Express 16, 8825-8834 (2008). [CrossRef] [PubMed]
- D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "A Composite medium with simultaneously negative permeability and permittivity," Phys. Rev. Lett. 84, 4184-4187 (2000). [CrossRef] [PubMed]
- N. Seddon and T. Bearpark, "Observation of the Inverse Doppler Effect," Science 302, 1537 (2003). [CrossRef] [PubMed]
- V. G. Veselago, "The electrodynamics of substances with simultaneously negative ∑ and μ," Soviet Physics Uspekhi 10, 509-514 (1964). [CrossRef]
- 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). http://www.sciencemag.org/cgi/reprint/314/5801/977.pdf. [CrossRef] [PubMed]
- I. Gil, J. G. Garcia, J. Bonache, F. Martin, M. Sorolla, and R. Marques, "Varactor-loaded split ring resonators for tunable notch filters at microwave frequencies," Electron. Lett. 40, 1347-1348 (2004). [CrossRef]
- S. Lim, C. Caloz, and T. Itoh, "Metamaterial-based electronically controlled transmission-line structure as a novel leaky-wave antenna with tunable radiation angle and beamwidth," IEEE Trans. Microwave Theory Tech. 52, 2678-2690 (2004). [CrossRef]
- T. Hand and S. Cummer, "Frequency Tunable Electromagnetic Metamaterial Using Ferroelectric Loaded Split Rings," J. Appl. Phys. 103, 066105 (2007). [CrossRef]
- H.-T. Chen, W. J. Padilla, J. M. O. Zide, A. C. Gossard, A. J. Tayor, and R. D. Averitt, "Active terahertz metamaterial devices," Nature 444, 597-600 (2006). [CrossRef] [PubMed]
- Q. Zhao, L. Kang, B. Du, B. Li, and J. Zhou, "Electrically tunable negative permeability metamaterials based on nematic liquid crystals," Appl. Phys. Lett. 90, 011112 (2007). [CrossRef]
- V. B. Bregar, "Effective-medium approach to the magnetic susceptibility of composites with ferromagnetic inclusions," Phys. Rev. B 71, 174418 (2005). [CrossRef]
- Y. He, P. He, S. D. Yoon, P. V. Parimi, F. J. Rachford, V. G. Harris, and C. Vittoria, "Tunable negative index metamaterial using yttrium iron garnet," J. Magn. Magn. Mater. 313, 187-191 (2007). [CrossRef]
- J. N. Gollub, D. R. Smith, and J. D. Baena, "Hybrid resonant phenomenon in a metamaterial structure with integrated resonant magnetic material," arXiv:0810.4871 (2008).
- J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, "Magnetism from conductors and enhanced nonlinear phenomena," IEEE Trans. Microwave Theory Tech. 47, 2075-2084 (1999). [CrossRef]
- R. Liu, T. J. Cui, D. Huang, B. Zhao, and D. R. Smith, "Description and explanation of electromagnetic behavior in artificial metamaterials based on effective medium theory," Phys. Rev. E 76, 026606 (2007). [CrossRef]
- 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]

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