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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 16 — Jul. 30, 2012
  • pp: 18377–18386
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Analysis of optical properties of metal/insulator/metal plasmonic metawaveguide with serial periodic stub resonators

Takeshi Baba  »View Author Affiliations


Optics Express, Vol. 20, Issue 16, pp. 18377-18386 (2012)
http://dx.doi.org/10.1364/OE.20.018377


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Abstract

The optical properties of the Au/Al2O3/Ag plasmonic metawaveguide with serial periodic stub resonators were numerically investigated. The refractive index of this design provides large value by using the ultra thin Al2O3 layer. Therefore, the dispersion relation which has negative group velocity, which is similar to the case of photonic crystals, can be formed in the sub diffraction energy range. The calculation results show the dependences of the negative group velocity dispersion curves on the size of the unit cell and the stub resonators. In addition, effective material properties are presented. From these analyses, it is found that this type of design has the property to strongly modulate the propagating characteristics of light.

© 2012 OSA

1. Introduction

Metamaterials are artificially structured materials whose electromagnetic properties are expressed in terms of effective material parameters. Those parameters are artificially designed to form the unit cell by conducting elements. Using the specific design of these elements, it is able to control the electromagnetic responses. Metamaterials have attractive electromagnetic properties such as negative refraction, super lensing and invisibility cloaking [1

1. R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292(5514), 77–79 (2001). [CrossRef] [PubMed]

4

4. 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]

].

The negative refraction can be realized in case that the effective permeability and permittivity are both negative. The most successful structure that has negative refractive index in the microwave region is a split ring resonator combined with metal wire structure. The split ring resonator shows negative permeability and metal wire provides negative permittivity. In recent years, with the scale down of the size of the unit cell, the various designs of thin layer structures, which have negative refractive index in optical wavelengths, were proposed such as cut-wire pair structure and double fish net structure [5

5. V. M. Shalaev, W. S. Cai, U. K. Chettiar, H. K. Yuan, A. K. Sarychev, V. P. Drachev, and A. V. Kildishev, “Negative index of refraction in optical metamaterials,” Opt. Lett. 30(24), 3356–3358 (2005). [CrossRef] [PubMed]

, 6

6. S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. Brueck, “Experimental demonstration of near-infrared negative-index metamaterials,” Phys. Rev. Lett. 95(13), 137404 (2005). [CrossRef] [PubMed]

]. Optical metamaterials which have negative refractive index provide potential application for optical elements, such as optical lithography, optical lenses and waveguides.

Negative refraction at visible frequencies has been directly observed in metal-insulator-metal (MIM) plasmonic waveguides [7

7. H. J. Lezec, J. Dionne, and H. Atwater, “Negative refraction at visible frequencies,” Science 316(5823), 430–432 (2007). [PubMed]

]. In addition, super lensing has been reported for plasmonic waveguides coated with a thin insulator layer (an insulator-insulator-metal, or IIM geometry). By reducing the thickness of insulator layer, a huge enhancement of refractive index of MIM plasmonic waveguides was reported at optical wavelengths [8

8. H. T. Miyazaki and Y. Kurokawa, “Controlled plasmon resonance in closed metal/insulator/metal nanocavities,” Appl. Phys. Lett. 89(21), 211126 (2006). [CrossRef]

]. For designing the optical metamaterials, these types of plasmonic waveguides are beneficial to make the structures in the wavelength which is below the diffraction limit.

2. The properties of Au/Al2O3/Ag plasmonic waveguide and stub resonator

The general properties of MIM plasmonic waveguide have been reported [12

12. E. N. Economou, “Surface plasmons in thin films,” Phys. Rev. 182(2), 539–554 (1969). [CrossRef]

15

15. G. Veronis and S. Fan, “Bends and splitters in metal-dielectric-metal subwavelength plasmonic waveguides,” Appl. Phys. Lett. 87(13), 131102 (2005). [CrossRef]

]. The refractive index of MIM plasmonic waveguide is much larger than the general semiconductor materials. The dispersion relations of Au (thickness 150 nm)/Al2O3 (thickness 10~120 nm)/Ag (thickness 150 nm) plasmonic waveguides were calculated by using FDTD simulation. The commercial software OPTIFDTD was used in our calculation [16]. In Fig. 1(a)
Fig. 1 (a) Schematic of simulation set up. Two observation points were positioned for calculating the phase difference and input pulse was set for x polarization. (b) Dispersion relation of the plasmonic waveguide for various Al2O3 layer thicknesses, whose values are described by the parameter t.
, the schematic of the calculation set up is shown. In FDTD simulations, the computational domain containing Yee cells with Δx = Δy = 2 or 5 nm were used and the whole region was surrounded by a perfect absorbing boundary. Plasmonic waveguides were set in the vacuum medium and input pulse was positioned in the Al2O3 center layer for x electric field polarization. The thicknesses of metal layers were set to be 150 nm. Thicknesses of insulator ware ranging from 10 nm to 120 nm. Reflection from the structure’s output interface is negligible because propagation modes are completely absorbed before getting to the structure’s output interface. Therefore, the observed values are not affected by the reflection from the structure’s interface. To calculate the dispersion relation of the waveguide mode, the two observation points inside the Al2O3 center layer were set and the phase differences (θ) of the y component of magnetic field between two observation points were derived. θ can be described by the real part of wave vector and the gap between two observation points. In Fig. 1(b), the calculated dispersion relations for various thicknesses of Al2O3 center layer (t) are shown.

Zstub=ZSZLiZStan(βstubd)ZSiZLtan(βstubd)
(1)

In Fig. 3(c), schematic of stub resonator and the calculation results of Zstub/ZMIM are shown. In this case, ZMIM stands for the impedance of the Au (thickness 150 nm)/Al2O3 (thickness 50 nm)/Ag (thickness 150 nm) plasmonic waveguide that doesn’t have stub resonator. It is confirmed that, with increase of the stub width, the resonance peak shows blue shifts due to the decreasing of the wave vector inside the stub. The wavelengths of the stub resonances can be determined from these calculations. In addition, the impedance of stub resonator goes higher at the resonance wavelength so that the reflection from the stub increases.

3. Analysis of MIM plasmonic waveguide with serial periodic stub resonators

The dispersion relation of Au (thickness 150 nm)/Al2O3 (thickness 50 nm)/Ag (thickness 150 nm) plasmonic waveguide with serial periodic stub resonators was also calculated by using FDTD simulation. By deriving the phase differences (θ) between two observation points at a distance of a, the real parts of the wave vector (kr) were calculated. And also, from the decrease of magnetic field amplitude (H) between observation points, the imaginary parts of the wave vector (ki) were calculated. For the calculations of kr and ki, these formulas were used:

k=kr+ikikr=θaki=ln|H(r+L)/H(r)|L
(2)

L is about 2 μm, which is adequate length to obtain appropriate value of ki. For calculation of kr, the range of θ was assumed from –π to + π considering 1st Brillouin zone (BZ). This assumption is valid because the waveguides have periodicity. Propagating modes are characterized by the dispersion relation inside the 1st BZ. In Fig. 4(a)
Fig. 4 (a)the design of the plasmonic waveguide with serial periodic stub resonators (b) Schematic of dispersion relation of plasmonic waveguide mode and stub resonance, where only positive group velocity dispersion curve is shown. (c) The calculation results of the dispersion relation for the condition of the size of the unit cell of 250 nm and the stub width of 30 nm. Only positive group velocity dispersion relation is shown. Near the 1.55 eV, the stub resonance is confirmed. (d) The imaginary part plot of the dispersion relation is presented. It is confirmed that the increases of the imaginary part of the wave vector are confirmed at the 1st BZ boundary, stub resonance energy and the Γ point.
and 4(b), the schematic of the waveguide and the dispersion relation of our system are shown. And, in Fig. 4(c), the calculation results of dispersion relation, which is limited to positive group velocity dispersion curve, are shown. In addition, in Fig. 4(d), the imaginary parts are shown.

As the size of the unit cell is set to be 250 nm, the wavelengths of negative group velocity region in Fig. 4(c) are longer than 500 nm, which is twice as long as the size of the unit cell. Therefore, the characterizing by using material parameters is meaning because there is no diffraction in those wavelength range. The stub resonance is confirmed at the 1.55 eV and its resonance couples to the waveguide mode which is folded at the 1st BZ boundary. In Fig. 4(d), the imaginary part of wave vector increases at the boundary of 1st BZ (near the 1.15 eV), the Γ point (near the 1.9 eV) and the stub resonance energy point. These results preserve the assumption that the phase differences between observation points ranges from –π to + π.

By reducing the stub-resonator width (w), shown in Fig. 5(a)
Fig. 5 The calculation results of the dispersion relations, where the size of the unit cell is 250 nm and the width of the stub is ranging from 30 nm to 80 nm. (a) The real part plot and (b) Imaginary part plot of the dispersion relations.
, these modes are shift towards the lower energy because the slope of the dispersion relation inclines. From Fig. 5(b), the imaginary part of wave vector increases at the boundary of 1st BZ (near the 1.15 eV), the Γ point (near the 1.9 eV) and the stub resonance wavelength (ranging 1.55 ~1.75 eV). For the 30 nm stub resonator width, however, the large increase of the imaginary part could not be confirmed clearly at the boundary of 1st BZ. This suggests the probability that the structure which has the stub width of 30 nm can’t be dealt as a periodic structure. Considering that the reflection from the stub goes weaker with the narrowing of the stub width and this waveguide is absorptive, periodicity of this waveguide doesn’t emerge in case that the reflection from the stub is too weak. Therefore, to investigate the condition that the band gap vanishes near the boundary of 1st BZ is needed. The dispersion relation for this waveguide can be written as Ref [9

9. A. Pannipitiya, I. D. Rukhlenko, M. Premaratne, H. T. Hattori, and G. P. Agrawal, “Improved transmission model for metal-dielectric-metal plasmonic waveguides with stub structure,” Opt. Express 18(6), 6191–6204 (2010). [CrossRef] [PubMed]

11

11. J. Liu, G. Fang, H. Zhao, Y. Zhang, and S. Liu, “Surface plasmon reflector based on serial stub structure,” Opt. Express 17(22), 20134–20139 (2009). [CrossRef] [PubMed]

]:

cos(ka)=2cos(βMIMa)[ZMIMZstub]sin(βMIMa)
(3)

k is a wave vector of Au/Al2O3/Ag plasmonic waveguide with serial periodic stub resonators, which can be expressed by using the data of dispersion calculation results in Fig. 5. ZMIM and βMIM means impedance and wave vector of the Au/Al2O3/Ag plasmonic waveguide. The value of |cos(ka)| was calculated. Generally, the band gap condition is|cos(ka)|>2, so that the evaluation of this value for various width of stub resonator provides the information whether this waveguide can be regarded as a periodic structure. In Fig. 6
Fig. 6 The calculation result of |cos(ka)|. Stub width ranges from 10 nm to 30 nm. Stub width ranges from 10 nm to 30 nm. in the case that the stub width is less than 20nm, the energy gap closes because the value is less than 2.
, the calculation results of |cos(ka)| for several stub widths ranging from 10 nm to 30 nm are shown.

4. The calculation of effective permittivity and permeability

The effective permittivity (εeff) and permeability (μeff) of Au (thickness 150 nm)/Al2O3 (thickness 50 nm)/Ag (thickness 150 nm) plasmonic waveguide with serial periodic stub resonators were investigated. The εeff and μeff provide the information whether the responses are magnetic or electric nature. Considering the calculation results of the dispersion relation which can be obtained in Fig. 4, it is worthwhile to study the dependence of those parameters on the size of the unit cell. To derive the effective permittivity and permeability, the calculation of the Bloch impedance (ZB) is needed because those parameters are expressed asεeff=n/ZB,μeff=nZB. In the periodic system, the impedance is expressed by Bloch impedance. In the transmission line theory, the Bloch impedance is expressed by using the voltage (V) and current (I) flow: ZB = V/I. The voltage and current correspond to the electric field which has the x polarization and magnetic field which has y polarization. For calculation, the electromagnetic fields obtained from one of the observation points were used, as is shown in Fig. 4(a). From these correspondences, the Bloch impedance was calculated. In Fig. 9
Fig. 9 (a)The calculation result of effective permittivity and (b)Effective permeability of Au/Al2O3 /Ag plasmonic waveguides with stub resonator. The sizes of the unit cell and stub width are 250 nm and 30 nm respectively.
, the effective permittivity and permeability calculation results are shown, where the size of the unit cell is 250 nm and stub width is 30 nm.

5. Conclusion

Acknowledgments

The author wishes to express his gratitude to Professor Teruya Ishihara and Professor Yoshiro Hirayama at Tohoku University for discussions.

References and links

1.

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292(5514), 77–79 (2001). [CrossRef] [PubMed]

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(18), 4184–4187 (2000). [CrossRef] [PubMed]

3.

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

4.

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]

5.

V. M. Shalaev, W. S. Cai, U. K. Chettiar, H. K. Yuan, A. K. Sarychev, V. P. Drachev, and A. V. Kildishev, “Negative index of refraction in optical metamaterials,” Opt. Lett. 30(24), 3356–3358 (2005). [CrossRef] [PubMed]

6.

S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. Brueck, “Experimental demonstration of near-infrared negative-index metamaterials,” Phys. Rev. Lett. 95(13), 137404 (2005). [CrossRef] [PubMed]

7.

H. J. Lezec, J. Dionne, and H. Atwater, “Negative refraction at visible frequencies,” Science 316(5823), 430–432 (2007). [PubMed]

8.

H. T. Miyazaki and Y. Kurokawa, “Controlled plasmon resonance in closed metal/insulator/metal nanocavities,” Appl. Phys. Lett. 89(21), 211126 (2006). [CrossRef]

9.

A. Pannipitiya, I. D. Rukhlenko, M. Premaratne, H. T. Hattori, and G. P. Agrawal, “Improved transmission model for metal-dielectric-metal plasmonic waveguides with stub structure,” Opt. Express 18(6), 6191–6204 (2010). [CrossRef] [PubMed]

10.

Y. Matsuzaki, T. Okamoto, M. Haraguchi, M. Fukui, and M. Nakagaki, “Characteristics of gap plasmon waveguide with stub structures,” Opt. Express 16(21), 16314–16325 (2008). [CrossRef] [PubMed]

11.

J. Liu, G. Fang, H. Zhao, Y. Zhang, and S. Liu, “Surface plasmon reflector based on serial stub structure,” Opt. Express 17(22), 20134–20139 (2009). [CrossRef] [PubMed]

12.

E. N. Economou, “Surface plasmons in thin films,” Phys. Rev. 182(2), 539–554 (1969). [CrossRef]

13.

J. A. Dionne, L. A. Sweatlock, A. Polman, and H. A. Atwater, “Plasmon slot waveguides: Towards chip-scale propagation with subwavelength-scale localization,” Phys. Rev. B 73(3), 035407–035416 (2006). [CrossRef]

14.

G. Veronis and S. Fan, “Modes of subwavelength plasmonic slot waveguides,” J. Lightwave Technol. 25(9), 2511–2521 (2007). [CrossRef]

15.

G. Veronis and S. Fan, “Bends and splitters in metal-dielectric-metal subwavelength plasmonic waveguides,” Appl. Phys. Lett. 87(13), 131102 (2005). [CrossRef]

16.

http://www.optiwave.com/

17.

D. M. Pozar, Microwave Engineering, 2nd ed. (Wiley, 1985).

OCIS Codes
(160.3918) Materials : Metamaterials
(350.4238) Other areas of optics : Nanophotonics and photonic crystals
(250.5403) Optoelectronics : Plasmonics

ToC Category:
Metamaterials

History
Original Manuscript: May 22, 2012
Revised Manuscript: July 1, 2012
Manuscript Accepted: July 13, 2012
Published: July 26, 2012

Citation
Takeshi Baba, "Analysis of optical properties of metal/insulator/metal plasmonic metawaveguide with serial periodic stub resonators," Opt. Express 20, 18377-18386 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-16-18377


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References

  1. R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science292(5514), 77–79 (2001). [CrossRef] [PubMed]
  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(18), 4184–4187 (2000). [CrossRef] [PubMed]
  3. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett.85(18), 3966–3969 (2000). [CrossRef] [PubMed]
  4. 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,” Science314(5801), 977–980 (2006). [CrossRef] [PubMed]
  5. V. M. Shalaev, W. S. Cai, U. K. Chettiar, H. K. Yuan, A. K. Sarychev, V. P. Drachev, and A. V. Kildishev, “Negative index of refraction in optical metamaterials,” Opt. Lett.30(24), 3356–3358 (2005). [CrossRef] [PubMed]
  6. S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. Brueck, “Experimental demonstration of near-infrared negative-index metamaterials,” Phys. Rev. Lett.95(13), 137404 (2005). [CrossRef] [PubMed]
  7. H. J. Lezec, J. Dionne, and H. Atwater, “Negative refraction at visible frequencies,” Science316(5823), 430–432 (2007). [PubMed]
  8. H. T. Miyazaki and Y. Kurokawa, “Controlled plasmon resonance in closed metal/insulator/metal nanocavities,” Appl. Phys. Lett.89(21), 211126 (2006). [CrossRef]
  9. A. Pannipitiya, I. D. Rukhlenko, M. Premaratne, H. T. Hattori, and G. P. Agrawal, “Improved transmission model for metal-dielectric-metal plasmonic waveguides with stub structure,” Opt. Express18(6), 6191–6204 (2010). [CrossRef] [PubMed]
  10. Y. Matsuzaki, T. Okamoto, M. Haraguchi, M. Fukui, and M. Nakagaki, “Characteristics of gap plasmon waveguide with stub structures,” Opt. Express16(21), 16314–16325 (2008). [CrossRef] [PubMed]
  11. J. Liu, G. Fang, H. Zhao, Y. Zhang, and S. Liu, “Surface plasmon reflector based on serial stub structure,” Opt. Express17(22), 20134–20139 (2009). [CrossRef] [PubMed]
  12. E. N. Economou, “Surface plasmons in thin films,” Phys. Rev.182(2), 539–554 (1969). [CrossRef]
  13. J. A. Dionne, L. A. Sweatlock, A. Polman, and H. A. Atwater, “Plasmon slot waveguides: Towards chip-scale propagation with subwavelength-scale localization,” Phys. Rev. B73(3), 035407–035416 (2006). [CrossRef]
  14. G. Veronis and S. Fan, “Modes of subwavelength plasmonic slot waveguides,” J. Lightwave Technol.25(9), 2511–2521 (2007). [CrossRef]
  15. G. Veronis and S. Fan, “Bends and splitters in metal-dielectric-metal subwavelength plasmonic waveguides,” Appl. Phys. Lett.87(13), 131102 (2005). [CrossRef]
  16. http://www.optiwave.com/
  17. D. M. Pozar, Microwave Engineering, 2nd ed. (Wiley, 1985).

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