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Journal of the Optical Society of America B

Journal of the Optical Society of America B

| OPTICAL PHYSICS

  • Editor: Grover Swartzlander
  • Vol. 30, Iss. 4 — Apr. 1, 2013
  • pp: 967–973
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Focusing property of the Au-Al2O3-Ag plasmonic metawaveguide with two-dimensional periodic stub resonators

Takeshi Baba  »View Author Affiliations


JOSA B, Vol. 30, Issue 4, pp. 967-973 (2013)
http://dx.doi.org/10.1364/JOSAB.30.000967


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Abstract

The optical properties of the Au / Al 2 O 3 / Ag plasmonic metawaveguide with two-dimensional periodic stub resonators were numerically investigated. The modes of this design can be characterized by the waveguide mode and stub resonance. The guided modes, which propagate inside the Al 2 O 3 layer, can be modulated by interacting with the stub resonators. In this situation, the negative group velocity dispersion relations in the near-infrared region can be realized. Focusing characteristics of this design were investigated by changing the design of the unit cell and the dispersion characteristics were discussed. In addition, the simplified design of the stub resonators was proposed. It was suggested that the proposed structure has the property to function as a nanotransmission line structure, which shows negative refraction at near infrared.

© 2013 Optical Society of America

1. INTRODUCTION

Metamaterials are artificially structured materials whose electromagnetic properties are expressed in terms of effective material parameters. Those parameters are artificially designed 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. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001). [CrossRef]

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, 977–980 (2006). [CrossRef]

].

In this work, the optical properties of the plasmonic metawaveguide with two-dimensional periodic stub resonators were numerically investigated by using finite-difference time-domain (FDTD) simulations. The results show the wavelength dependence of the dispersion curves and focusing properties of this two-dimensional system. It can be considered that these results present another design of the nanotransmission line which shows negative refraction because via was removed in this design [16

16. C. Caloz and T. Itoh, Electromagnetic Metamaterials (Wiley, 2006).

,17

17. G. V. Eleftheriades and K. G. Balmain, Negative-Refraction Metamaterials (Wiley, 2005).

]. In Section 2, the dispersion calculation results of the Au/Al2O3/Ag plasmonic metawaveguide with serial periodic stub resonators are shown for various sizes of the unit cell were shown. In Section 3, the focusing properties of the Au/Al2O3/Ag plasmonic metawaveguide were presented.

2. DISPERSION CURVES OF Au/Al2O3/Ag PLASMONIC WAVEGUIDE WITH TWO-DIMENSIONAL PERIODIC STUB RESONATORS

The general design of MIM plasmonic waveguide with stub resonators, which has previously reported in [11

11. T. Baba, “Analysis of optical properties of metal/insulator/metal plasmonic metawaveguides with serial stub resonators,” Opt. Express 20, 18377–18386 (2012). [CrossRef]

14

14. L. Yang, C. Min, and G. Veronis, “Guided subwavelength slow-light mode supported by a plasmonic waveguide system,” Opt. Lett. 35, 4184–4186 (2010). [CrossRef]

], is considered to be the structure with grooves inside the waveguide. Although the stub resonator was capped by the metal layer, this design can be simplified by eliminating the metal cap. In Fig. 1, the simplified design of the plasmonic waveguide with two-dimensional stub resonators is presented.

Fig. 1. Schematic of the plasmonic waveguide with two-dimensional periodic stub resonators. The pitch and the width of the stub resonator correspond to a and w, respectively. The gap between the Au dots behaves as a stub resonator.

The Au dots are aligned for the square-lattice structure on the Al2O3/Ag layer. The gaps between the Au dots, this behaves as a stub resonator, were filled with Al2O3. The medium above the Au dots is air. In this design, the Au and Ag metals were used considering the convention of the fabricating process. a corresponds to the pitch of the Au dots and the w stands for the width of the stub resonator. The thickness of the Au dots is 150 nm. The thicknesses of the Al2O3 center layer and Ag layer are 50 and 150 nm, respectively. Eliminating the metal cap layer, however, is not a crucial because the stub resonance exists in this system. The dispersion relations for ΓX were calculated by using FDTD simulation. The commercial software OPTIFDTD was used in this calculation [18]. In Fig. 2, the schematics of the simulation setup are shown. In FDTD simulations, the computational domain containing Yee cells with Δx=Δy=Δz=10nm were used and the whole region was surrounded by a periodic and perfect absorbing boundaries. This cell size is relatively rough so that the refractive index of the waveguide mode is different from the calculation results of smaller grid size. However, this difference does not provide the essential change of the property of the waveguide due to the little difference of the value of waveguide mode. The structure was contacted to the periodic boundary for y-direction. Plasmonic waveguides were set in the vacuum medium and input pulse was positioned in the Al2O3 center layer for z electric field polarization. Reflection from the structure’s output interface is negligible because the absorption loss of this design is as high as at least 100 dB per 10 μm so that propagation modes are completely absorbed before reaching the output interface. Therefore, the observed values are not affected by the reflection from the structure’s output 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 multiplying the real part of wave vector and the gap between two observation points. By deriving θ between two observation points at a distance of a, the real parts of the wave vector (kr) were calculated. For calculation of kr, the range of θ was assumed from π to +π considering first BZ. This assumption is valid because the waveguides have periodicity.

Fig. 2. (a) Schematic of the FDTD simulation set up for ΓX dispersion calculation. The periodic boundary condition was set for the boundary of the y-axis and the perfect absorbing boundary condition was set for the boundary of z- and x-axis. The structure was contacted by the periodic boundary for y-direction. (b) Simulation setup of the observation points, where these were positioned under the Au dots. The distance between these points is a.

Propagating modes are characterized by the dispersion relation inside the first BZ. In Fig. 3, the calculation results of dispersion relation for ΓX, which is limited to positive group velocity dispersion curve, is presented, where the w is 30 nm and a is 250 nm. These parameters were chosen in the same way for [11

11. T. Baba, “Analysis of optical properties of metal/insulator/metal plasmonic metawaveguides with serial stub resonators,” Opt. Express 20, 18377–18386 (2012). [CrossRef]

].

Fig. 3. Calculation result of the dispersion relation for ΓX, where the a and w are 250 and 30 nm, respectively. Only positive group velocity dispersion relation is presented.

In Fig. 3, the dispersion relation provides the positive and negative value of the wave vector because the periodic structure can be characterized by the folded-zone scheme. The energy range, shown in Fig. 3, is lower than the energy of the diffraction limit. It is confirmed that the negative group velocity dispersion exists in the energy range from 1 to 1.8 eV. The wave vector, which has positive value, exists near 1.2 eV because the propagating mode inside the waveguide couples to the outside mode via stub resonator so that the wave vector changes its sign. In Fig. 4, the magnetic amplitude distributions of y-direction (Hy) were presented for various energies.

Fig. 4. Magnetic field amplitude distributions for various energies of input light (a) 1.4 eV, (b) 1.2 eV, (c) 1.05 eV.

From the results in Fig. 4, it was confirmed that the waveguide mode has negative sign of wave vector localized inside the Al2O3 layer and, on the other hand, the amplitude distribution at 1.2 eV spreads outside the waveguide so that the wave vector has the positive sign. By regarding this structure as a transmission line network, the dispersion relation can be derived [16

16. C. Caloz and T. Itoh, Electromagnetic Metamaterials (Wiley, 2006).

19

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

]. The formula of the dispersion relation is as follows:
(1eikxa)2eikxa+(1eikya)2eikyaZY=0.
(1)
The parameter ZY means the multiplication of the series impedance (Z) and shunt admittance (Y). kx and ky stand for the wave vector for each direction. ZY can be extracted by using the numerical result of the dispersion relation for ΓX, which was obtained in Fig. 3. Although it is well understood that the plasmonic effect drastically changes the property of microwave-transmission lines, the ZY is the any value. Therefore, by extracting the value of ZY from dispersion relation for ΓX, the plasmonic effect can be included. From the above formula, the equivalent energy surface of this system can be derived.

In Fig. 5, the equivalent energy surfaces for various energies are shown. The equivalent energy surfaces form a diamond-like shape. Therefore, the refractive index of this waveguide has anisotropy. On the other hand, refractive index modulation by stub resonance increases with reducing the size of the unit cell as mentioned in [11

11. T. Baba, “Analysis of optical properties of metal/insulator/metal plasmonic metawaveguides with serial stub resonators,” Opt. Express 20, 18377–18386 (2012). [CrossRef]

] because the stub resonators couple to each other. In the condition that a and w are 100 and 20 nm, respectively, the sharp wavelength dependence was confirmed at the stub-resonance wavelength. This tendency is same in this two-dimensional periodic system. In Fig. 6, the dispersion relations of this two-dimensional periodic system are presented. The sharp response of the stub resonator was confirmed and the energy of stub resonance can be controlled by changing the height of the stub resonator (thickness of Au dot). Reducing the height of the stub, the resonance energy shifts toward the shorter wavelength because the Fabry–Perot resonance inside the stub has its tendency in general case. In the case that the thickness of Au dots is 100 nm, the resonance was confirmed near 1 eV. Although the dispersion has noisy curve below 1 eV, this is due to the artifact of the simulation due to wide spreading of the waveguide mode in to the vacuum medium. The equivalent energy surface can be derived by using the above formula (1) and the result is presented at the stub resonance energy in Fig. 7. The circle shape can be confirmed so that the structure behaves as isotropic material at the stub resonance energy. This tendency is quite different from the condition that a=250nm.

Fig. 5. Equivalent energy surfaces of the plasmonic waveguide with two-dimensional periodic stub resonators for various energies.
Fig. 6. Dispersion relation for ΓX. a and w are 100 and 20 nm, respectively. (a) The thickness of Au dots are 150 nm and (b) the thickness of Au dots are 100 nm.
Fig. 7. Equivalent energy surface near the plasmonic metawaveguide with two-dimensional periodic stub resonators near the 1 eV, where a and w are 100 and 20 nm, respectively. The thickness of Au dots is 100 nm.

3. FOCUSING PROPERTIES OF AU/AL2O3/AG PLASMONIC WAVEGUIDE WITH TWO-DIMENSIONAL PERIODIC STUB RESONATORS

Next, the focusing properties of this system were investigated. In Fig. 8, the simulation setup is shown. The three periods of this periodic structure were set for x-direction and the length of the structure for y-direction is about 5 μm. The whole structure was surrounded by the perfect absorbing boundary and the grid was set to be 10 nm. As shown in Fig. 8, the point source was set inside the Au/Al2O3/Ag waveguide. d corresponds to the distance between point source and the structure’s input interface. The electromagnetic distribution was monitored inside the Al2O3 center layer. In Fig. 9, the amplitude distributions of magnetic field are shown for various input energies, where a=250nm, d=0.3μm, and w=30nm. The calculation results show the input-energy dependence of the flat lens. While the input energy above 1.03 eV, the focusing effect was confirmed. As for the input energy of 0.83 eV, the waveguide mode was not focused because the sign of refractive index was positive. At the 1.65 eV, whose value is near the Γ point, the focusing was unclear because the absolute value of refractive index is very low. On the other hand, at the 1.38 and 1.03 eV, clear focusing was obtained. As for the 1.03 eV, however, there were some focusing points. This is due to the diffraction nature of this focusing effect so that these spots were constructed by the stub near the output interface.

Fig. 8. Simulation setup for the focusing analysis. (a) Side view and (b) top view.
Fig. 9. Simulation results of negative refractive flat lens for various input energies. The amplitude distributions of magnetic field are presented, where a=250nm, d=0.3μm, and w=30nm. The energies of liput light were (a) 1.65 eV, (b) 1.38 eV, (c) 1.03 eV, (d) 0.83 eV.

In Fig. 10, the amplitude distributions of magnetic field were shown, where a=250nm and w=30nm. The input energies were 1.38 and 1.03 eV. In addition, the dependence of the focusing properties on d is presented.

Fig. 10. Simulation results of negative refractive flat lens for various positions of point source. The amplitude distributions of magnetic field are presented, where a=250nm and w=30nm. The energies of liput light were (a) 1.38 eV and (b) 1.03 eV.

In condition that a=100nm and w=20nm, the sharp resonance of the stub was confirmed and the refractive index shows negative sign in that energy. The focusing property of this stub resonance was investigated in the same way. In Fig. 11, the simulation results are shown, where the height of the stub resonator was 100 nm and three periods were set for x-direction and the length of the structure for y-direction is 5 μm.

Fig. 11. Simulation results of negative refractive flat lens for various positions of point source. The amplitude distributions of magnetic field are presented, where a=100nm and w=20nm and the energy of input light was nearly 1 eV. (a) d=0.1μm, (b) d=0.3μm, (c) d=0.5μm.

The output light was focused in the conditions of d=0.1 and 0.3 μm. For d=0.5μm, however, the focusing of output light could not be confirmed because d is larger than the length of the waveguide. These results are clearer than the case of Fig. 10 because the equivalent energy surface shows circle shape and the propagating modes inside the periodic structure mainly localize in the stub. As for the dependence on the d, however, was not confirmed clearly because it was considered that the loss of this mode was large. This tendency is similar to the case of pointing vector distribution. In Fig. 12, the distributions of the magnitude of the pointing vector are shown.

Fig. 12. Magnitude distributions of pointing vector are presented, where a=100nm and w=20nm and the energy of input light was nearly 1 eV. (a) d=0.1μm, (b) d=0.3μm, and (c) d=0.5μm.

Although the focusing points of the pointing vector have slightly different positions from the results of magnetic field distributions, the dependence on d is almost the same. This difference is because the proposed design has strong evanescent components. From these results, the focusing effect that is originated from the negative refraction was confirmed. In this simulation, the simulation area is relatively small, which corresponds to 3 by 5 μm. However, it was confirmed that the same results could be obtained by using broader simulation area.

4. CONCLUSION

In this work, the Au/Al2O3/Ag plasmonic metawaveguide with two-dimensional periodic stub resonators was simplified by eliminating the cap layer of the metal. In addition, the dispersion and focusing properties were investigated for various designs of the unit cell. In the condition that a=250nm and w=30nm, the focusing effect of the flat lens was observed and its equivalent energy surface shows anisotropic. In case that a=100nm and w=20nm, the focusing was confirmed, which was originated from the stub resonance. In this condition, the equivalent energy surface shows isotropic and the focusing property of the structure was strongly modulated by the stub resonance. It was found that incorporating the stub resonator inside the Au/Al2O3/Ag plasmonic waveguide is an effective way to modulate the guiding property. This simple design is quite different from the general transmission line metamaterial, which shows negative refraction because this proposed design does not need to connect the metal layer via [17

17. G. V. Eleftheriades and K. G. Balmain, Negative-Refraction Metamaterials (Wiley, 2005).

,19

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

]. Absorption loss in this design, however, is as large as at least 100 dB per 10 μm, which was indicated in [11

11. T. Baba, “Analysis of optical properties of metal/insulator/metal plasmonic metawaveguides with serial stub resonators,” Opt. Express 20, 18377–18386 (2012). [CrossRef]

]. Therefore, to solve this problem it is necessary to use the loss less metal. It can be considered that the stub resonance exists in this case.

REFERENCES

1.

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

2.

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

3.

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

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, 977–980 (2006). [CrossRef]

5.

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

6.

J. A. Dionne, E. Verhagen, A. Polman, and H. Atwater, “Are negative index materials achievable with surface plasmon waveguides? A case study of three-plasmonic geometries,” Opt. Express 16, 19001–19017 (2008). [CrossRef]

7.

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

8.

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

9.

C. Min and G. Veronis, “Absorption switches in metal-dielectric-metal plasmonic waveguides,” Opt. Express 17, 10757–10766 (2009). [CrossRef]

10.

A. V. Krasavin and A. Zayats, “Silicon-based plasmonic waveguides,” Opt. Express 18, 11791–11799 (2010). [CrossRef]

11.

T. Baba, “Analysis of optical properties of metal/insulator/metal plasmonic metawaveguides with serial stub resonators,” Opt. Express 20, 18377–18386 (2012). [CrossRef]

12.

A. Pannipitiya, I. D. Rukhlenko, M. Premarantne, H. T. Hattori, and G. P. Agrawal, “Improved transmission model for metal-dielectric-metal plasmonic waveguides with stub structure,” Opt. Express 18, 6191–6203 (2010). [CrossRef]

13.

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

14.

L. Yang, C. Min, and G. Veronis, “Guided subwavelength slow-light mode supported by a plasmonic waveguide system,” Opt. Lett. 35, 4184–4186 (2010). [CrossRef]

15.

M. Notomi, “Negative refraction in photonic crystals,” Opt. Quantum Electron. 34, 133–143 (2002). [CrossRef]

16.

C. Caloz and T. Itoh, Electromagnetic Metamaterials (Wiley, 2006).

17.

G. V. Eleftheriades and K. G. Balmain, Negative-Refraction Metamaterials (Wiley, 2005).

18.

http://www.optiwave.com/.

19.

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:
Optoelectronics

History
Original Manuscript: November 2, 2012
Revised Manuscript: January 13, 2013
Manuscript Accepted: February 18, 2013
Published: March 18, 2013

Citation
Takeshi Baba, "Focusing property of the Au-Al2O3-Ag plasmonic metawaveguide with two-dimensional periodic stub resonators," J. Opt. Soc. Am. B 30, 967-973 (2013)
http://www.opticsinfobase.org/josab/abstract.cfm?URI=josab-30-4-967


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References

  1. R. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001). [CrossRef]
  2. D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schults, “A composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000). [CrossRef]
  3. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000). [CrossRef]
  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, 977–980 (2006). [CrossRef]
  5. E. N. Economou, “Surface plasmons in thin films,” Phys. Rev. 182, 539–554 (1969). [CrossRef]
  6. J. A. Dionne, E. Verhagen, A. Polman, and H. Atwater, “Are negative index materials achievable with surface plasmon waveguides? A case study of three-plasmonic geometries,” Opt. Express 16, 19001–19017 (2008). [CrossRef]
  7. H. J. Lezec, J. Dionne, and H. Atwater, “Negative refraction at visible frequencies,” Science 316, 430–432 (2007). [CrossRef]
  8. H. T. Miyazaki and Y. Kurokawa, “Controlled plasmon resonance in closed metal/insulator/metal nanocavities,” Appl. Phys. Lett. 89, 211126 (2006). [CrossRef]
  9. C. Min, and G. Veronis, “Absorption switches in metal-dielectric-metal plasmonic waveguides,” Opt. Express 17, 10757–10766 (2009). [CrossRef]
  10. A. V. Krasavin, and A. Zayats, “Silicon-based plasmonic waveguides,” Opt. Express 18, 11791–11799 (2010). [CrossRef]
  11. T. Baba, “Analysis of optical properties of metal/insulator/metal plasmonic metawaveguides with serial stub resonators,” Opt. Express 20, 18377–18386 (2012). [CrossRef]
  12. A. Pannipitiya, I. D. Rukhlenko, M. Premarantne, H. T. Hattori, and G. P. Agrawal, “Improved transmission model for metal-dielectric-metal plasmonic waveguides with stub structure,” Opt. Express 18, 6191–6203 (2010). [CrossRef]
  13. J. Liu, G. Fang, H. Zhao, Y. Zhang, and S. Liu, “Surface plasmon reflector based on serial stub structure,” Opt. Express 17, 20134–20139 (2009). [CrossRef]
  14. L. Yang, C. Min, and G. Veronis, “Guided subwavelength slow-light mode supported by a plasmonic waveguide system,” Opt. Lett. 35, 4184–4186 (2010). [CrossRef]
  15. M. Notomi, “Negative refraction in photonic crystals,” Opt. Quantum Electron. 34, 133–143 (2002). [CrossRef]
  16. C. Caloz, and T. Itoh, Electromagnetic Metamaterials (Wiley, 2006).
  17. G. V. Eleftheriades and K. G. Balmain, Negative-Refraction Metamaterials (Wiley, 2005).
  18. http://www.optiwave.com/ .
  19. D. M. Pozar, Microwave Engineering, 2nd ed. (Wiley, 1985).

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