## Measurement of surface plasmon correlation length differences using Fibonacci deterministic hole arrays |

Optics Express, Vol. 20, Issue 14, pp. 15222-15231 (2012)

http://dx.doi.org/10.1364/OE.20.015222

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### Abstract

Using terahertz (THz) transmission measurements through two-dimensional Fibonacci deterministic subwavelength hole arrays fabricated in metal foils, we find that the surface plasmon-polariton (SPP) correlation lengths for aperiodic resonances are smaller than those associated with the underlying grid. The enhanced transmission spectra associated with these arrays contain two groups of Fano-type resonances: those related to the two-dimensional Fibonacci structure and those related to the underlying hole grid array upon which the aperiodic Fibonacci array is built. For both groups the destructive interference frequencies at which transmission minima occur closely match prominent reciprocal vectors in the hole array (HA) structure-factor in reciprocal space. However the Fibonacci-related transmission resonances are much weaker than both their calculated Fourier intensity in **k** space and the grid-related resonances. These differences may arise from the complex, multi-fractal dispersion relations and scattering from the underlying grid arrays. We also systematically studied and compared the transmission resonance strength of Fibonacci HA and periodic HA lattices as a function of the number of holes in the array structure. We found that the Fibonacci-related resonance strengths are an order of magnitude weaker than that of the periodic HA, consistent with the smaller SPP correlation length for the aperiodic structure.

© 2012 OSA

## 1. Introduction

6. M. I. Stockman, S. V. Faleev, and D. J. Bergman, “Localization versus delocalization of surface plasmons in nanosystems: Can one state have both characteristics?” Phys. Rev. Lett. **87**(16), 167401 (2001). [CrossRef] [PubMed]

*increases with disorder*, and has an ‘inverse Matthiessen rule’, where it

*increases with the temperature*[7

7. D. Mayou, C. Berger, F. Cyrot-Lackmann, T. Klein, and P. Lanco, “Evidence for unconventional electronic transport in quasicrystals,” Phys. Rev. Lett. **70**(25), 3915–3918 (1993). [CrossRef] [PubMed]

8. M. Kohmoto, B. Sutherland, and K. Iguchi, “Localization of optics: Quasiperiodic media,” Phys. Rev. Lett. **58**(23), 2436–2438 (1987). [CrossRef] [PubMed]

10. R. W. Peng, X. Q. Huang, F. Qiu, M. Wang, A. Hu, S. S. Jiang, and M. Mazzer, “Symmetry-induced perfect transmission of light waves in quasiperiodic dielectric multilayers,” Appl. Phys. Lett. **80**(17), 3063–3065 (2002). [CrossRef]

*et al*. recently reported near-field optical microscopy measurements on two-dimensional (2D) Fibonacci nano-particle structures [11

11. R. Dallapiccola, A. Gopinath, F. Stellacci, and L. Dal Negro, “Quasi-periodic distribution of plasmon modes in two-dimensional Fibonacci arrays of metal nanoparticles,” Opt. Express **16**(8), 5544–5555 (2008). [CrossRef] [PubMed]

12. A. Gopinath, S. V. Boriskina, B. M. Reinhard, and L. Dal Negro, “Deterministic aperiodic arrays of metal nanoparticles for surface-enhanced Raman scattering (SERS),” Opt. Express **17**(5), 3741–3753 (2009). [CrossRef] [PubMed]

## 2. Experimental details

_{j + 1}= {S

_{j-1}, S

_{j}} for j ≥1, where S

_{0}= {B}, S

_{1}= {A} and S

_{j}is a structure obtained after

*j*iterations of the generation rule; here A and B are seed letters. 2D Fibonacci structures have been developed previously by several groups [13

13. R. Lifshitz, “The square Fibonacci tiling,” J. Alloy. Comp. **342**(1-2), 186–190 (2002). [CrossRef]

15. L. D. Negro, N. Feng, and A. Gopinath, “Electromagnetic coupling and plasmon localization in deterministic aperiodic arrays,” J. Opt. A, Pure Appl. Opt. **10**(6), 064013 (2008). [CrossRef]

15. L. D. Negro, N. Feng, and A. Gopinath, “Electromagnetic coupling and plasmon localization in deterministic aperiodic arrays,” J. Opt. A, Pure Appl. Opt. **10**(6), 064013 (2008). [CrossRef]

_{x}: A→AB, B→A and g

_{y}: A→B, B→BA in two orthogonal directions (x,y). We designed and fabricated HA structures with circular apertures on 5x5 cm

^{2}area of 75 µm thick free-standing stainless steel foils using this generation method, where A is an aperture, and B is a space lacking an aperture (i.e. a missing hole). The details of the HA generation method can be found elsewhere [11

11. R. Dallapiccola, A. Gopinath, F. Stellacci, and L. Dal Negro, “Quasi-periodic distribution of plasmon modes in two-dimensional Fibonacci arrays of metal nanoparticles,” Opt. Express **16**(8), 5544–5555 (2008). [CrossRef] [PubMed]

*grid*HA that also contributes to the structure factor. We fabricated HA patterns of periodic, 2D Fibonacci, and disordered structures with various numbers of holes in arrays of up to 90x90, with varying hole diameters using a computerized generation scheme.

*magnitude and phase*, t(ω) in the range ~0.1 THz to 0.5 THz. The resulting Fourier transformed data may be described by the relation:

_{incident}and E

_{transmitted}are the incident and transmitted THz electric fields, respectively, and |t(ω)| and φ(ω) are the magnitude and phase of the amplitude transmission coefficient, respectively. The THz-TDS technique is unique in that it allows for a direct measurement of the transient THz electric field transmitted through the structures, yielding both amplitude and phase information. From these spectra both real and imaginary components of the dielectric response, ε(ω) can be directly obtained without the need for Kramers-Kronig transformations, where somewhat arbitrary assumptions about asymptotic behavior are typically made.

## 3. Experimental results and discussion

*randomly generated*on a square grid having the same number of holes and nearest neighbor distance,

*a*as the 2D Fibonacci HA structures; these structures are denoted ‘random on grid’, or RG. Figure 1(c) shows an example of such an RG aperture structure in which 173 holes are randomly placed on a square grid with 0.8 mm spacing. The 2D FFT spectrum of this RG aperture structure contains only two sets of the

**G**reciprocal vectors (

**G**and

_{1}**G**respectively) that arise from the grid structure, with no other prominent reciprocal vector visible; this confirms that the aperture distribution in this sample is indeed random.

_{2},*d*[4

4. T. Matsui, A. Agrawal, A. Nahata, and Z. V. Vardeny, “Transmission resonances through aperiodic arrays of subwavelength apertures,” Nature **446**(7135), 517–521 (2007). [CrossRef] [PubMed]

*a*but different

*d*, ranging from 0.3 mm to 0.68 mm. These diameters correspond to the aperture waveguide cut-off frequency,

*f*

_{c}ranging from 0.58 THz to 0.26 THz, respectively [18

18. 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*strengthens the F-resonances at frequencies close to

*f*

_{c}, since the Fano resonance coupling between a discrete EOT feature and the transmission continuum due to the individual holes becomes stronger when the discrete frequency is close to

*f*

_{c}[16

16. U. Fano, “Effects of configuration interaction on intensities and phase shifts,” Phys. Rev. **124**(6), 1866–1878 (1961). [CrossRef]

17. A. Miroshnichenko, S. Flach, and Y. Kivshar, “Fano resonances in nanoscale structures,” Rev. Mod. Phys. **82**(3), 2257–2298 (2010). [CrossRef]

*ε*(

*ω*) of the Fibonacci HA from the transmission amplitude and phase (not shown) given in Fig. 2(a) (

*d*= 0.68 mm). We have previously shown that

*ε*(

*ω*) can be well described by a superposition of a non-resonant (continuous) and resonant (discrete) components given by the formula proposed by Agrawal

*et al*. [18

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

*ε*(

*ω*) of periodic HAs [18

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

*ε*(

*ω*) of Fibonacci HAs is much more complicated because of the multitude of resonances associated with the

**G**-vectors as well as the

**F**-vectors. We note the enhanced intensity of the F

_{2}resonance close to

*f*

_{c}of the individual holes. We also note that the F-resonance at 0.15 THz more clearly appears in this HA structure in agreement with our finding in the t

_{F}/t

_{RG}spectrum of Fig. 2(b).

*N*, while keeping constant the lattice constant

*a*= 0.8 mm and diameter

*d*= 0.44 mm; and periodic HA having the same number of holes,

*d*, and

*a*. Figure 3(a) shows t(ω) of a Fibonacci HA structure with 800 holes; the characteristic G- and F-resonances are clearly seen. The spectrum t(ω) of the corresponding random HA having the same

*N*and

*d*is also shown in Fig. 3(a). The F

_{2}-resonance is chosen for studying the resonance intensity,

*I*, defined as the area under the transmission of the F

_{2}peak with respect to the transmission of the corresponding random HA structure. The F

_{2}resonance intensities defined in this way are plotted vs.

*N*in Fig. 3(b). We also calculated the resonance peak intensity from the FT intensity associated with the corresponding reciprocal vectors in the structure factor; Fig. 3(b)

*inset*demonstrates this calculation for the FT F

_{2}intensity. We found that the intensity of both resonances increases nonlinearly with

*N.*In addition it is clear that the experimental intensities are smaller than the calculated intensities from the FT. For the G

_{1}-resonance we need a multiplication factor of ~4 to match the experimental to the calculated intensities. A factor of 2 is justified because the polarized THz beam that subtends only two G-vectors out of possible four; another factor of two shows that in reality there are some SPP losses in the HA structure. However for the F

_{2}resonance we need a factor of 24(!) to match the theoretical calculation. We therefore conclude that the F-resonances are

*much weaker*than the G-resonances in the Fibonacci HA structure, and do not fit the FT calculation even relative to the G

_{1}resonance. This shows that the F-resonances are

*inherently*weak, and it therefore requires a more sophisticated explanation.

*a*= 0.8 mm and hole diameter

*d*= 0.44 mm. As in the case of Fibonacci HAs, we extracted the intensity of transmission resonance by integrating the area limited by the G

_{1}peak envelope and the corresponding random HA, as shown in Fig. 3(c). The periodic G

_{1}transmission intensity dependence on the number of holes is shown in Fig. 3(d). We also calculated the transmission strength of the peak G

_{1}follow the same method as described above for the Fibonacci HA. We notice that the FFT spectrum was calculated from arrays of points located in the centers of holes. Therefore the structure factor in Figs. 1(b) and 1(d) does not contain the contribution of the continuum spectrum, which is due to the transmission through individual holes in the experiment. This is the reason that we compare the theory with the experimental transmission strength extracted from the area difference between the transmission spectra of the Fibonacci (or periodic) HAs and their corresponding random HAs. Interestingly, the experimental data shown in Figs. 3(c) and 3(d) follow the calculation trends very well. We note that the periodic resonance strength substantially increases with the number of holes, N, for relatively small values of N; but start to saturate for values of N that are smaller than the value of N for which saturation take place with the Fibonacci F

_{2}transmission strength.

_{1}transmission strength is nearly two orders of magnitude stronger than the Fibonacci F

_{2}transmission strength, and the Fibonacci experimental strength is an order of magnitude smaller than its calculation value. A possible reason that might lead to this large difference between the G- and F-resonant strength is the proximity of the different resonances to the cut-off frequency,

*f*

_{c}. While the G

_{1}resonance frequency at 0.375 THz is close to

*f*

_{c}~0.4 THz, the F

_{2}resonance frequency at 0.27 THz is further away (see Fig. 2(c)). The strongest transmission strength for the F

_{2}resonance occurs for HA structures having

*d*= 0.68 mm, which is ~5 times stronger than it is for structures with

*d*= 0.44 mm. Nevertheless, the transmission strength F

_{2}at

*d*= 0.68 mm is still an order of magnitude weaker than the transmission strength of the G

_{1}resonance at

*d*= 0.44 mm. We therefore conclude that the G-resonances are much stronger than the F-resonances even if the proximity to the cut-off frequency

*f*

_{c}is taken into account. This is in contrast with the behavior of hot electromagnetic spot intensity in Fibonacci structures using nano-particle arrays [11

11. R. Dallapiccola, A. Gopinath, F. Stellacci, and L. Dal Negro, “Quasi-periodic distribution of plasmon modes in two-dimensional Fibonacci arrays of metal nanoparticles,” Opt. Express **16**(8), 5544–5555 (2008). [CrossRef] [PubMed]

^{B}

**(**

_{k}**r**)~exp(-i

**k**•

**r**); so that the ‘attenuated’ Bloch function in real materials is Ψ

^{at}

**(**

_{k}**r**)~exp(-i

**k**•

**r**)exp(-r/R), where the second exponential decay term describes the amplitude attenuation with the distance, r. To see the effect of the ‘attenuation term’ on the strength of the reciprocal vectors in

**k**-space due to smaller interference, we calculated the structure factor of a periodic square HA using Ψ

^{at}

**(**

_{k}**r**) rather than Ψ

^{B}

**(**

_{k}**r**), as depicted in Fig. 4 . It is clearly seen that the intensity of the FT peak above the background diminishes substantially at small R (Fig. 4(a)). To further study this effect we calculated the integrated intensity of the G

_{1}transmission resonance in a periodic HA as a function of R (Fig. 4(b)). It is seen that the integrated intensity of the transmission G

_{1}-resonance increases quadratic with R.

*et al*. [19

19. F. Przybilla, C. Genet, and T. W. Ebbesen, “Long vs. short-range orders in random subwavelength hole arrays,” Opt. Express **20**(4), 4697–4709 (2012). [CrossRef] [PubMed]

19. F. Przybilla, C. Genet, and T. W. Ebbesen, “Long vs. short-range orders in random subwavelength hole arrays,” Opt. Express **20**(4), 4697–4709 (2012). [CrossRef] [PubMed]

## 4. Conclusion

## Acknowledgments

## References and links

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

2. | H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science |

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

4. | T. Matsui, A. Agrawal, A. Nahata, and Z. V. Vardeny, “Transmission resonances through aperiodic arrays of subwavelength apertures,” Nature |

5. | A. Agrawal, T. Matsui, W. Zhu, A. Nahata, and Z. V. Vardeny, “Terahertz spectroscopy of plasmonic fractals,” Phys. Rev. Lett. |

6. | M. I. Stockman, S. V. Faleev, and D. J. Bergman, “Localization versus delocalization of surface plasmons in nanosystems: Can one state have both characteristics?” Phys. Rev. Lett. |

7. | D. Mayou, C. Berger, F. Cyrot-Lackmann, T. Klein, and P. Lanco, “Evidence for unconventional electronic transport in quasicrystals,” Phys. Rev. Lett. |

8. | M. Kohmoto, B. Sutherland, and K. Iguchi, “Localization of optics: Quasiperiodic media,” Phys. Rev. Lett. |

9. | W. Gellermann, M. Kohmoto, B. Sutherland, and P. C. Taylor, “Localization of light waves in Fibonacci dielectric multilayers,” Phys. Rev. Lett. |

10. | R. W. Peng, X. Q. Huang, F. Qiu, M. Wang, A. Hu, S. S. Jiang, and M. Mazzer, “Symmetry-induced perfect transmission of light waves in quasiperiodic dielectric multilayers,” Appl. Phys. Lett. |

11. | R. Dallapiccola, A. Gopinath, F. Stellacci, and L. Dal Negro, “Quasi-periodic distribution of plasmon modes in two-dimensional Fibonacci arrays of metal nanoparticles,” Opt. Express |

12. | A. Gopinath, S. V. Boriskina, B. M. Reinhard, and L. Dal Negro, “Deterministic aperiodic arrays of metal nanoparticles for surface-enhanced Raman scattering (SERS),” Opt. Express |

13. | R. Lifshitz, “The square Fibonacci tiling,” J. Alloy. Comp. |

14. | X. Fu, Y. Liu, B. Cheng, and D. Zheng, “Spectral structure of two-dimensional Fibonacci quasilattices,” Phys. Rev. B Condens. Matter |

15. | L. D. Negro, N. Feng, and A. Gopinath, “Electromagnetic coupling and plasmon localization in deterministic aperiodic arrays,” J. Opt. A, Pure Appl. Opt. |

16. | U. Fano, “Effects of configuration interaction on intensities and phase shifts,” Phys. Rev. |

17. | A. Miroshnichenko, S. Flach, and Y. Kivshar, “Fano resonances in nanoscale structures,” Rev. Mod. Phys. |

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

19. | F. Przybilla, C. Genet, and T. W. Ebbesen, “Long vs. short-range orders in random subwavelength hole arrays,” Opt. Express |

**OCIS Codes**

(050.1220) Diffraction and gratings : Apertures

(240.6680) Optics at surfaces : Surface plasmons

(260.3090) Physical optics : Infrared, far

(160.1245) Materials : Artificially engineered materials

**ToC Category:**

Optics at Surfaces

**History**

Original Manuscript: March 14, 2012

Revised Manuscript: June 13, 2012

Manuscript Accepted: June 14, 2012

Published: June 22, 2012

**Citation**

Tho Duc Nguyen, Ajay Nahata, and Z. Valy Vardeny, "Measurement of surface plasmon correlation length differences using Fibonacci deterministic hole arrays," Opt. Express **20**, 15222-15231 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-14-15222

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### References

- T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature391(6668), 667–669 (1998). [CrossRef]
- H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science297(5582), 820–822 (2002). [CrossRef] [PubMed]
- J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science305(5685), 847–848 (2004). [CrossRef] [PubMed]
- T. Matsui, A. Agrawal, A. Nahata, and Z. V. Vardeny, “Transmission resonances through aperiodic arrays of subwavelength apertures,” Nature446(7135), 517–521 (2007). [CrossRef] [PubMed]
- A. Agrawal, T. Matsui, W. Zhu, A. Nahata, and Z. V. Vardeny, “Terahertz spectroscopy of plasmonic fractals,” Phys. Rev. Lett.102(11), 113901 (2009). [CrossRef] [PubMed]
- M. I. Stockman, S. V. Faleev, and D. J. Bergman, “Localization versus delocalization of surface plasmons in nanosystems: Can one state have both characteristics?” Phys. Rev. Lett.87(16), 167401 (2001). [CrossRef] [PubMed]
- D. Mayou, C. Berger, F. Cyrot-Lackmann, T. Klein, and P. Lanco, “Evidence for unconventional electronic transport in quasicrystals,” Phys. Rev. Lett.70(25), 3915–3918 (1993). [CrossRef] [PubMed]
- M. Kohmoto, B. Sutherland, and K. Iguchi, “Localization of optics: Quasiperiodic media,” Phys. Rev. Lett.58(23), 2436–2438 (1987). [CrossRef] [PubMed]
- W. Gellermann, M. Kohmoto, B. Sutherland, and P. C. Taylor, “Localization of light waves in Fibonacci dielectric multilayers,” Phys. Rev. Lett.72(5), 633–636 (1994). [CrossRef] [PubMed]
- R. W. Peng, X. Q. Huang, F. Qiu, M. Wang, A. Hu, S. S. Jiang, and M. Mazzer, “Symmetry-induced perfect transmission of light waves in quasiperiodic dielectric multilayers,” Appl. Phys. Lett.80(17), 3063–3065 (2002). [CrossRef]
- R. Dallapiccola, A. Gopinath, F. Stellacci, and L. Dal Negro, “Quasi-periodic distribution of plasmon modes in two-dimensional Fibonacci arrays of metal nanoparticles,” Opt. Express16(8), 5544–5555 (2008). [CrossRef] [PubMed]
- A. Gopinath, S. V. Boriskina, B. M. Reinhard, and L. Dal Negro, “Deterministic aperiodic arrays of metal nanoparticles for surface-enhanced Raman scattering (SERS),” Opt. Express17(5), 3741–3753 (2009). [CrossRef] [PubMed]
- R. Lifshitz, “The square Fibonacci tiling,” J. Alloy. Comp.342(1-2), 186–190 (2002). [CrossRef]
- X. Fu, Y. Liu, B. Cheng, and D. Zheng, “Spectral structure of two-dimensional Fibonacci quasilattices,” Phys. Rev. B Condens. Matter43(13), 10808–10814 (1991). [CrossRef] [PubMed]
- L. D. Negro, N. Feng, and A. Gopinath, “Electromagnetic coupling and plasmon localization in deterministic aperiodic arrays,” J. Opt. A, Pure Appl. Opt.10(6), 064013 (2008). [CrossRef]
- U. Fano, “Effects of configuration interaction on intensities and phase shifts,” Phys. Rev.124(6), 1866–1878 (1961). [CrossRef]
- A. Miroshnichenko, S. Flach, and Y. Kivshar, “Fano resonances in nanoscale structures,” Rev. Mod. Phys.82(3), 2257–2298 (2010). [CrossRef]
- A. Agrawal, Z. V. Vardeny, and A. Nahata, “Engineering the dielectric function of plasmonic lattices,” Opt. Express16(13), 9601–9613 (2008). [CrossRef] [PubMed]
- F. Przybilla, C. Genet, and T. W. Ebbesen, “Long vs. short-range orders in random subwavelength hole arrays,” Opt. Express20(4), 4697–4709 (2012). [CrossRef] [PubMed]

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