## Resonant infrared transmission and effective medium response of subwavelength H-fractal apertures

Optics Express, Vol. 18, Issue 4, pp. 3946-3951 (2010)

http://dx.doi.org/10.1364/OE.18.003946

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### Abstract

The transmission through periodic arrays of subwavelength H-fractal apertures in a gold film at infrared wavelengths is investigated numerically. H-fractal apertures support subwavelength cut-off resonances that are hybridized with surface plasmons along the sidewalls of the aperture. Enhanced transmission occurs at wavelengths that are about ten times the aperture side length. The highly subwavelength size scale of the H-fractal enables an effective medium parameter description for the aperture array, which reveals a lossy plasma permittivity and a diamagnetic permeability.

© 2010 OSA

## 1. Introduction

1. W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature **424**(6950), 824–830 (2003). [CrossRef] [PubMed]

2. R. Gordon, A. G. Brolo, D. Sinton, and K. L. Kavanagh, “Resonant optical transmission through hole-arrays in metal films: physics and applications,” Laser Photon. Rev. 1–25 (2009) (**DOI** ). [CrossRef]

*ω*, of most metals, a periodic array of subwavelength apertures in a thin metal film can excite

_{p}*extended*surface plasmons (SPs) that mediate the optical transmission. This grating-induced transmission resonance was observed by Ebbesen et al. in a square lattice of circular holes with radii ~1/10 of the wavelength [3

3. T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through subwavelength hole arrays,” Nature **391**(6668), 667–669 (1998). [CrossRef]

4. C. Genet and T. W. Ebbesen, “Light in tiny holes,” Nature **445**(7123), 39–46 (2007). [CrossRef] [PubMed]

*localized*subwavelength resonances, which depends solely on the shape of the single aperture [5

5. H. Cao and A. Nahata, “Influence of aperture shape on the transmission properties of a periodic array of subwavelength apertures,” Opt. Express **12**(16), 3664–3672 (2004). [CrossRef] [PubMed]

7. Z. Ruan and M. Qiu, “Enhanced transmission through periodic arrays of subwavelength holes: the role of localized waveguide resonances,” Phys. Rev. Lett. **96**(23), 233901 (2006). [CrossRef] [PubMed]

8. C. Huang, Q. Wang, and Y. Zhu, “Dual effect of surface plasmons in light transmission through perforated metal films,” Phys. Rev. B **75**(24), 245421 (2007). [CrossRef]

9. A. Mary, S. G. Rodrigo, L. Martin-Moreno, and F. J. Garcia-Vidal, “Theory of light transmission through an array of rectangular holes,” Phys. Rev. B **76**(19), 195414 (2007). [CrossRef]

11. W. Wen, L. Zhou, B. Hou, C. T. Chan, and P. Sheng, “Resonant transmission of microwave through subwavelength fractal slits in a metallic plate,” Phys. Rev. B **72**(15), 153406 (2005). [CrossRef]

12. Y. Poujet, J. Salvi, and F. I. Baida, “90% Extraordinary optical transmission in the visible range through annular aperture metallic arrays,” Opt. Lett. **32**(20), 2942–2944 (2007). [CrossRef] [PubMed]

14. L. Sun and L. Hesselink, “Low-loss subwavelength metal C-aperture waveguide,” Opt. Lett. **31**(24), 3606–3608 (2006). [CrossRef] [PubMed]

*ε*(

^{eff}*ω*), of the lossy plasma form, and an effective permeability,

*μ*(

^{eff}*ω*), which is diamagnetic.

## 2. H-Fractal aperture

*d*. The “H” is then scaled to half of the original size, i.e. each side is of length

*d*/2. The four half-sized “H”s are referred as the “second order” of the fractal pattern, and their centers are at the end points of the previous “H.” The side length of the fractal pattern is

*s*. Continuing this scaling and pattern generation procedure indefinitely leads to a fractal which fills a

*w*, is neglected.

## 3. Enhanced infrared transmission and cut-off resonance

*a*= 600 nm in a 100 nm thick gold film, which exhibits a peak of ~70% at 72.5 THz (wavelength

*λ*= 4.13 μm), corresponding to an intensity transmission that is more than twice the aperture area fraction of 21%. The peak wavelength is ~8.6 times the side length of the aperture,

*s = 480 nm*. The power flow is concentrated in the central slit of the H-fractal pattern [Fig. 2(b)]. Consequently, the magnitude of the electric field inside the slit is enhanced by about 25 times compared to incident E-field [Fig. 2(c)]. Also shown in Fig. 2(a) is the transmission of the simple H-shaped apertures without the set of four smaller H-shapes in the second order fractal. The transmission peak is at 132 THz and the spatial pattern of the power flow is similar to Fig. 1(b). The addition of higher orders in the fractal pattern can down-shift the fundamental resonance frequency by introducing finer spatial features to the resonant mode, which increases

*λ/s*further. Although high order fractal patterns can support other resonances which are localized away from the central slit [11

11. W. Wen, L. Zhou, B. Hou, C. T. Chan, and P. Sheng, “Resonant transmission of microwave through subwavelength fractal slits in a metallic plate,” Phys. Rev. B **72**(15), 153406 (2005). [CrossRef]

*β*, and the phase accumulation through the aperture are close to zero. The reduced transmitted amplitude for increasing film thickness is due to losses in the metal.

*β*= 0 occurs at the cut-off wavelengths for a loss-less waveguide, the resonance frequency of the H-fractal apertures can be interpreted in terms of the cut-off wavelength of an equivalent aperture waveguide. The dispersion relation for the fundamental waveguide mode of the H-fractal aperture, assuming it is infinitely long in the

*z*-direction, is shown in Fig. 4(a) .

*β*is normalized to

*k*=

_{0}*ω/c*, where

*c*is the speed of light in vacuum. Periodic boundary conditions are imposed along

*x*and

*y*with the Bloch wavevector set to

*K*0. At 64.5 THz (λ = 4.65 μm), Im[

_{x}= K_{y}=*β*] = Re[

*β*], and the corresponding waveguide mode is essentially identical to the finite thickness resonance mode in terms of the magnitude of the Poynting vector (Fig. 4(b) and Fig. 2(b)), as well as the EM field. For a lossy waveguide, the cut-off condition is not unambiguous as in a lossless waveguide. Here, we define “cut-off” as Im[

*β*] = Re[

*β*], which represents a properly attenuated wave. The transmission peak frequency is higher than the cut-off frequency of 64.5 THz, due to the finite thickness of the film. As the film thickness increases, the peak frequency approaches the cut-off frequency. Thus, the resonance of the finite thickness apertures can be referred to as a “cut-off resonance.”

*E*component of the real metal and PEC waveguide modes, since it is parallel to the metal boundary and is a feature of SPs. Figures 4(c) and (d) show at Re(

_{z}*β /k*) = 0.35,

_{0}*E*is non-zero and localized at the air-metal interface for the real metal waveguide mode, and is zero in the PEC case. As in other aperture waveguides [16

_{z}16. R. Gordon and A. G. Brolo, “Increased cut-off wavelength for a subwavelength hole in a real metal,” Opt. Express **13**(6), 1933–1938 (2005). [CrossRef] [PubMed]

18. S. Collin, F. Pardo, and J. L. Pelouard, “Waveguiding in nanoscale metallic apertures,” Opt. Express **15**(7), 4310–4320 (2007). [CrossRef] [PubMed]

## 4. Effective medium parameters

*ε*, and effective permeability,

^{eff}*μ*, description. The dispersion relations of the H-fractal waveguide and bulk gold (inset of Fig. 4(a)) illustrate that the cut-off frequency of the H-fractal waveguide has a similar role as the plasma frequency in metals. Within the PEC approximation, Pendry et al. have shown that a periodic array of square holes in a semi-infinite metal layer has an effective

^{eff}*ω*given by the cut-off frequency of the square hole waveguide and an effective diamagnetic response [19

_{p}19. J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science **305**(5685), 847–848 (2004). [CrossRef] [PubMed]

20. F. J. Garcia-Vidal, L. Martin-Moreno, and J. B. Pendry, “Surfaces with holes in them: new plasmonic metamaterials,” J. Opt. A, Pure Appl. Opt. **7**(2), S97–S101 (2005). [CrossRef]

*ω*, the hole waveguide cut-off wavelength should be much larger than the hole size and the periodicity, requiring the holes to be filled with high index dielectrics. Otherwise, the periodicity becomes comparable to the wavelength, and effective parameters are not well defined [21

_{p}21. A. Agrawal, Z. V. Vardeny, and A. Nahata, “Engineering the dielectric function of plasmonic lattices,” Opt. Express **16**(13), 9601–9613 (2008). [CrossRef] [PubMed]

22. 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**(19), 195104 (2002). [CrossRef]

*ω*(i.e., metal refractive index is ~0). The retrieved effective parameters are shown in Fig. 5(b) and are specific to the polarization due to the structural anisotropy of the fractal pattern.

_{p}*ε*is of the lossy plasma form where Re(

_{x}^{eff}*ε*) is zero at the transmission peak frequency, and

_{x}^{eff}*μ*is diamagnetic with Re(

_{y}^{eff}*μ*) ≈0.23 at low frequencies.

_{y}^{eff}*et al*. [19

19. J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science **305**(5685), 847–848 (2004). [CrossRef] [PubMed]

*A*is a constant,

*ω*is the new plasma frequency from the aperture geometry, and the nonzero

_{p}^{eff}*γ*accounts for the losses in the real metal. We determine

^{eff}*A*= 37.7 and

*γ*= 0.0688

^{eff}*ω*from the retrieved effective permittivity at 30 THz, where

_{p}^{eff}*λ/a*is the largest, and

*ω*is the peak frequency of 66.5 THz. The results are shown as open squares in Fig. 5(b). The

_{p}^{eff}*ε*obtained from Eq. (2) agrees well with the directly retrieved values (from low frequencies to

_{x}^{eff}*ω*). The complicated behavior of

_{p}^{eff}*ε*and Im(

_{x}^{eff}*μ*) < 0 at ω >

_{y}^{eff}*ω*illustrates the limited validity of the effective parameters at decreasing values of

_{p}^{eff}*λ/a*[23].

## 5. Conclusions

*ε*is of the plasma form and

^{eff}*μ*is diamagnetic. The transmission and reflection properties of the metallic fractal apertures are controllable by the aperture geometry, such as slit length and scaling orders, and the excitation of SPs in the narrowly-spaced aperture walls. The strength of the interaction between the SPs on the sidewalls depends on the line width of the aperture. A wider line width will upshift the resonance frequency due to the diminished interaction of the SPs. Therefore, by adding higher orders to a fractal pattern and/or modifying the line width, an effective plasma permittivity can be created and tuned over a large frequency range beyond the constraints imposed by the intrinsic properties of metals or dielectrics. These effects can enable a greater flexibility in the design of SP structures.

^{eff}## Acknowledgements

## References and links

1. | W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature |

2. | R. Gordon, A. G. Brolo, D. Sinton, and K. L. Kavanagh, “Resonant optical transmission through hole-arrays in metal films: physics and applications,” Laser Photon. Rev. 1–25 (2009) ( |

3. | T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through subwavelength hole arrays,” Nature |

4. | C. Genet and T. W. Ebbesen, “Light in tiny holes,” Nature |

5. | H. Cao and A. Nahata, “Influence of aperture shape on the transmission properties of a periodic array of subwavelength apertures,” Opt. Express |

6. | K. J. van der Molen, K. J. Klein Koerkamp, S. Enoch, F. B. Segerink, N. F. van Hulst, and L. Kuiperts, “Role of shape and localized resonances in extraordinary transmission through periodic arrays of subwavelength holes: Experiment and theory,” Phys. Rev. B |

7. | Z. Ruan and M. Qiu, “Enhanced transmission through periodic arrays of subwavelength holes: the role of localized waveguide resonances,” Phys. Rev. Lett. |

8. | C. Huang, Q. Wang, and Y. Zhu, “Dual effect of surface plasmons in light transmission through perforated metal films,” Phys. Rev. B |

9. | A. Mary, S. G. Rodrigo, L. Martin-Moreno, and F. J. Garcia-Vidal, “Theory of light transmission through an array of rectangular holes,” Phys. Rev. B |

10. | D. H. Werner, and R. Mittra, |

11. | W. Wen, L. Zhou, B. Hou, C. T. Chan, and P. Sheng, “Resonant transmission of microwave through subwavelength fractal slits in a metallic plate,” Phys. Rev. B |

12. | Y. Poujet, J. Salvi, and F. I. Baida, “90% Extraordinary optical transmission in the visible range through annular aperture metallic arrays,” Opt. Lett. |

13. | L. Lin, L. B. Hande, and A. Roberts, “Resonant nanometric cross-shaped apertures: Single apertures versus periodic arrays,” Appl. Phys. Lett. |

14. | L. Sun and L. Hesselink, “Low-loss subwavelength metal C-aperture waveguide,” Opt. Lett. |

15. | E. D. Palik, |

16. | R. Gordon and A. G. Brolo, “Increased cut-off wavelength for a subwavelength hole in a real metal,” Opt. Express |

17. | H. Shin, P. B. Catrysse, and S. Fan, “Effect of the plasmonic dispersion relation on the transmission properties of subwavelength cylindrical holes,” Phys. Rev. B |

18. | S. Collin, F. Pardo, and J. L. Pelouard, “Waveguiding in nanoscale metallic apertures,” Opt. Express |

19. | J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science |

20. | F. J. Garcia-Vidal, L. Martin-Moreno, and J. B. Pendry, “Surfaces with holes in them: new plasmonic metamaterials,” J. Opt. A, Pure Appl. Opt. |

21. | A. Agrawal, Z. V. Vardeny, and A. Nahata, “Engineering the dielectric function of plasmonic lattices,” Opt. Express |

22. | 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 |

23. | T. Koschny, P. Markos, D. R. Smith, and C. M. Soukoulis, “Resonant and antiresonant frequency dependence of the effective parameters of metamaterials,” Phys. Rev. |

**OCIS Codes**

(160.3918) Materials : Metamaterials

(250.5403) Optoelectronics : Plasmonics

(310.6628) Thin films : Subwavelength structures, nanostructures

**ToC Category:**

Metamaterials

**History**

Original Manuscript: December 14, 2009

Revised Manuscript: January 25, 2010

Manuscript Accepted: February 4, 2010

Published: February 12, 2010

**Citation**

Bo Hou, Xin Qing Liao, and Joyce K. S. Poon, "Resonant infrared transmission and effective medium response of subwavelength H-fractal apertures," Opt. Express **18**, 3946-3951 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-4-3946

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### References

- W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003). [CrossRef] [PubMed]
- R. Gordon, A. G. Brolo, D. Sinton, and K. L. Kavanagh, “Resonant optical transmission through hole-arrays in metal films: physics and applications,” Laser Photon. Rev. 1–25 (2009) (DOI ). [CrossRef]
- T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through subwavelength hole arrays,” Nature 391(6668), 667–669 (1998). [CrossRef]
- C. Genet and T. W. Ebbesen, “Light in tiny holes,” Nature 445(7123), 39–46 (2007). [CrossRef] [PubMed]
- H. Cao and A. Nahata, “Influence of aperture shape on the transmission properties of a periodic array of subwavelength apertures,” Opt. Express 12(16), 3664–3672 (2004). [CrossRef] [PubMed]
- K. J. van der Molen, K. J. Klein Koerkamp, S. Enoch, F. B. Segerink, N. F. van Hulst, and L. Kuiperts, “Role of shape and localized resonances in extraordinary transmission through periodic arrays of subwavelength holes: Experiment and theory,” Phys. Rev. B 72(4), 045421 (2005). [CrossRef]
- Z. Ruan and M. Qiu, “Enhanced transmission through periodic arrays of subwavelength holes: the role of localized waveguide resonances,” Phys. Rev. Lett. 96(23), 233901 (2006). [CrossRef] [PubMed]
- C. Huang, Q. Wang, and Y. Zhu, “Dual effect of surface plasmons in light transmission through perforated metal films,” Phys. Rev. B 75(24), 245421 (2007). [CrossRef]
- A. Mary, S. G. Rodrigo, L. Martin-Moreno, and F. J. Garcia-Vidal, “Theory of light transmission through an array of rectangular holes,” Phys. Rev. B 76(19), 195414 (2007). [CrossRef]
- D. H. Werner, and R. Mittra, Frontiers in Electromagnetics (IEEE Press, Piscataway, NJ, 2000), Chap. 2.
- W. Wen, L. Zhou, B. Hou, C. T. Chan, and P. Sheng, “Resonant transmission of microwave through subwavelength fractal slits in a metallic plate,” Phys. Rev. B 72(15), 153406 (2005). [CrossRef]
- Y. Poujet, J. Salvi, and F. I. Baida, “90% Extraordinary optical transmission in the visible range through annular aperture metallic arrays,” Opt. Lett. 32(20), 2942–2944 (2007). [CrossRef] [PubMed]
- L. Lin, L. B. Hande, and A. Roberts, “Resonant nanometric cross-shaped apertures: Single apertures versus periodic arrays,” Appl. Phys. Lett. 95(20), 201116 (2009). [CrossRef]
- L. Sun and L. Hesselink, “Low-loss subwavelength metal C-aperture waveguide,” Opt. Lett. 31(24), 3606–3608 (2006). [CrossRef] [PubMed]
- E. D. Palik, Handbook of optical constants of solids (Academic Press, Orlando, FL, 1985).
- R. Gordon and A. G. Brolo, “Increased cut-off wavelength for a subwavelength hole in a real metal,” Opt. Express 13(6), 1933–1938 (2005). [CrossRef] [PubMed]
- H. Shin, P. B. Catrysse, and S. Fan, “Effect of the plasmonic dispersion relation on the transmission properties of subwavelength cylindrical holes,” Phys. Rev. B 72(8), 085436 (2005). [CrossRef]
- S. Collin, F. Pardo, and J. L. Pelouard, “Waveguiding in nanoscale metallic apertures,” Opt. Express 15(7), 4310–4320 (2007). [CrossRef] [PubMed]
- J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science 305(5685), 847–848 (2004). [CrossRef] [PubMed]
- F. J. Garcia-Vidal, L. Martin-Moreno, and J. B. Pendry, “Surfaces with holes in them: new plasmonic metamaterials,” J. Opt. A, Pure Appl. Opt. 7(2), S97–S101 (2005). [CrossRef]
- A. Agrawal, Z. V. Vardeny, and A. Nahata, “Engineering the dielectric function of plasmonic lattices,” Opt. Express 16(13), 9601–9613 (2008). [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(19), 195104 (2002). [CrossRef]
- T. Koschny, P. Markos, D. R. Smith, and C. M. Soukoulis, “Resonant and antiresonant frequency dependence of the effective parameters of metamaterials,” Phys. Rev. 68, 065602 (2003).

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