## Plasmon-polariton nano-strip resonators: from visible to infra-red

Optics Express, Vol. 16, Issue 10, pp. 6867-6876 (2008)

http://dx.doi.org/10.1364/OE.16.006867

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### Abstract

Dispersion of the resonant properties exhibited by silver and gold nano-strips in a wide range of wavelengths is considered. The tunability and Q-factor of scattering resonances as well as the field enhancement achieved at strip terminations are analyzed in the wavelength range from visible to near infrared (400–1700 nm), confirming that the resonant behaviour is dominated by dispersion properties of short-range surface-plasmon polaritons (SR-SPPs) propagating along the strip. It is found that, while the Q-factor decreases for longer wavelengths due to the SR-SPP dispersion curve moving closer to the light line, the field enhancement depending also on the metal susceptibility magnitude remains largely unaffected. The results obtained are also used to estimate the phase change involved in the SR-SPP reflection by strip terminations.

© 2008 Optical Society of America

## 1. Introduction

2. S. Lal, S. Link, and N. J. Halas, “Nano-optics from sensing to waveguiding,” Nat. Photonics **1**, 641–648 (2007). [CrossRef]

3. F. Wang and Y. R. Shen, “General properties of local plasmons in metal nanostructures,” Phys. Rev. Lett. **97**, 206806-1–4 (2006). [CrossRef] [PubMed]

*electrostatic*resonances. Different particle shapes, such as spheres, triangles, cubes, and nano-shells have been thoroughly investigated (see Ref. [2

2. S. Lal, S. Link, and N. J. Halas, “Nano-optics from sensing to waveguiding,” Nat. Photonics **1**, 641–648 (2007). [CrossRef]

14. T. Søndergaard, “Modeling of plasmonic nanostructures: Green’s function integral equation methods,” Phys. Status Solidi (b) **244**, 3448–3462 (2007). [CrossRef]

15. D. W. Prather, M. S. Mirotznik, and J. N. Mait, “Boundary integral methods applied to the analysis of diffractive optical elements,” J. Opt. Soc. Am. A **14**, 34–43 (1997). [CrossRef]

## 2. Spectral properties of short-range SPPs in thin metal films

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

*t*(i.e. two metal-dielectric interfaces displaced by a distance

*t*) two SPP super-modes exist: a short-range SPP (SR-SPP) and a long-range SPP (LR-SPP) mode. This nomenclature accounts for the dramatic difference in propagation length (namely the propagation distance after which the plasmonic field is reduced by a 1/

*e*factor), which results to be in the range of tens of

*µ*m and few mm for SR-SPP and LR-SPP respectively. Another nomenclature is also adopted in the literature, namely

*slow*-SPP and

*fast*-SPP modes respectively, accounting for another remarkable difference: phase velocity. In fact, it is well known that SR-SPPs are more tightly confined to the metal-dielectric interface, with a significant field component inside the metal (in a sense, SR-SPPs are more

*electronic*in nature), with a stronger coupling of the field to the oscillatory motion of conductive electrons. This accounts not only for large propagation losses but also causes a significant slowing in the propagation speed, thus an increase in the effective mode index.

18. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B **6**, 4370–4379 (1972). [CrossRef]

*n*decreases from 3.46 at 390 nm to 1.07 at 1550 nm. On the contrary, the propagation length scales more than linearly with wavelength, and results to be about four times longer in silver with respect to gold, as expected in view of lower silver conduction losses. Note also that thinner films results in slower (higher effective index) and more lossy (shorter propagation length) SR-SPP modes.

_{eff}*n*=

_{eff}*Re*{

*k*/

_{srsp}*k*

_{0}} and

*L*=[ 2

*Im*{

*k*}]

_{srsp}^{-1}respectively, where

*k*

_{0}=2

*π*/

*λ*and

*k*is the SR-SPP propagation constant given by the analytical formula [12

_{srsp}12. S. I. Bozhevolnyi and T. Søndergaard, “General properties of slow-plasmon resonant nanostructures: nanoantennas and resonators,” Opt. Express **15**, 10869–10877 (2007). [CrossRef] [PubMed]

12. S. I. Bozhevolnyi and T. Søndergaard, “General properties of slow-plasmon resonant nanostructures: nanoantennas and resonators,” Opt. Express **15**, 10869–10877 (2007). [CrossRef] [PubMed]

*n*≃-2/(

_{eff}*Re*{

*ε*}

_{m}*k*

_{0}

*t*), where

*ε*is the metal dielectric constant. Considering that, for noble metals, the real part of the dielectric constant increases in magnitude almost quadratically with increasing wavelength, the previous formula well synthesizes the dispersion properties of SR-SPP noticed above.

_{m}## 3. Nano-strip resonators based on SR-SPP modes

*t*and width

*w*surrounded by air (ideally vacuum). The strip length along the z-axis was assumed to be much longer than

*w*(ideally infinite), thus allowing a rigorous 2D-modelling in the

*xy*plane. The structure was excited by an incident

*p*-polarized plane wave propagating at an angle of 45° with respect to the

*x*axis, and the strip scattering cross section as a function of wavelength was computed. Fig. 2(b) shows samples of these scattering spectra for three silver strips of 10 nm thickness. Note several peaks emerging from the background, clearly revealing the intrinsic resonant behaviour of these structures.

10. T. Søndergaard and S. I. Bozhevolnyi, “Slow-plasmon resonant nanostructures: Scattering and field enhancements,” Phys. Rev. B **75**, 073402-1–4 (2007). [CrossRef]

11. T. Søndergaard and S. I. Bozhevolnyi, “Metal nano-strip optical resonators,” Opt. Express **15**, 4198–4204 (2007). [CrossRef] [PubMed]

12. S. I. Bozhevolnyi and T. Søndergaard, “General properties of slow-plasmon resonant nanostructures: nanoantennas and resonators,” Opt. Express **15**, 10869–10877 (2007). [CrossRef] [PubMed]

*n*is the effective index of a SR-SPP mode bound to and propagating along a metal film with the same thickness as the strip,

_{eff}*m*=1,2,3,… accounts for the order of the resonance, and

*ϕ*is a phase change (modulus

*π*) due to reflection at the edges. Typically, in very short strips only the first order resonance is sustained and a single peak appears in the scattering cross-section, as for the 100 nm wide silver strip of Fig. 2(b) exhibiting a quite sharp resonance at 535 nm. On the contrary, longer structures exhibit resonances of different order. As an example, the 300-nm wide strip of Fig. 2(b) shows two resonant peaks in the scattering spectrum, one at 968 nm and another at 589 nm. Note that, in any case, the first order resonant wavelength is much longer than twice the strip width. This particular behaviour is precisely due to the high effective index exhibited by SR-SPP modes, which causes Eq. (2) to be fulfilled for wavelengths much longer than the geometrical size.

## 4. Broad-band wavelength tunability of nano-strip resonators

*ϕ*, which is also expected to be wavelength-dependent, is identified. Unfortunately, an evaluation of this parameter starting from first principles is quite complicated and still lacking. Furthermore, the accuracy of Eq. (2) for very short (

*w*<100 nm) and very long (

*w*~1

*µ*m) nano-strips has been never checked.

*w*in a huge range of values, comprised between 40 nm to 1200 nm, for silver and gold nanostrips of thickness

*t*=5, 10, 15 nm. For each width

*w*the scattering cross-section was calculated similar to Fig. 2 and first and second order resonance wavelengths were identified with 0.5 nm accuracy. Results are collected in Fig. 3 (for silver) and Fig. 4 (for gold) and presented according to a design point of view by showing the strip width as a function of the desired resonant wavelength.

8. L. Novotny, “Effective wavelength scaling for optical antennas,” Phys. Rev. Lett. **98**, 266802-1–4 (2007). [CrossRef] [PubMed]

9. J. Takahara, S. Yamagishi, H. Taki, A. Morimoto, and T. Kobayashi, “Guiding of a one-dimensional optical beam with nanometer diameter,” Opt. Lett. **22**, 475–477 (1997). [CrossRef] [PubMed]

**15**, 10869–10877 (2007). [CrossRef] [PubMed]

8. L. Novotny, “Effective wavelength scaling for optical antennas,” Phys. Rev. Lett. **98**, 266802-1–4 (2007). [CrossRef] [PubMed]

*n*≃-2/(

_{eff}*Re*{

*ε*}

_{m}*k*

_{0}

*t*) instead of Eq. (1). In fact, according to this approximation, a factor of 2 decrease in the slope of the tuning curve is expected by moving from 10 nm to 5 nm thick structures. On the contrary, Fig. 3 and Fig. 4 clearly show that the slope decrease is actually quite small.

## 5. Q-factor and field enhancement

### 5.1. Q-factor

*Q*=

*λ*/

_{P}*FWHM*, where

*λ*is the peak wavelength and

_{P}*FWHM*is the Full Width at Half Maximum of the resonance in the scattering spectrum [see Fig. 2(b)].

18. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B **6**, 4370–4379 (1972). [CrossRef]

### 5.2. Field Enhancement under resonant excitation

10. T. Søndergaard and S. I. Bozhevolnyi, “Slow-plasmon resonant nanostructures: Scattering and field enhancements,” Phys. Rev. B **75**, 073402-1–4 (2007). [CrossRef]

11. T. Søndergaard and S. I. Bozhevolnyi, “Metal nano-strip optical resonators,” Opt. Express **15**, 4198–4204 (2007). [CrossRef] [PubMed]

**15**, 10869–10877 (2007). [CrossRef] [PubMed]

13. T. Søndergaard, J. Beermann, A. Boltasseva, and S. I. Bozhevolnyi, “Slow-plasmon resonant nano-strip antennas: Analysis and demonstration,” Phys. Rev. B **77**, 115420-1–5 (2008). [CrossRef]

**E**/

**E**

_{0}| with

**E**

_{0}being the electric field of the incident plane wave) at the maximum of the standing wave pattern of the electric field inside the resonator (point M in the inset of Fig. 6). Though this point is actually not experimentally accessible, nevertheless it is interesting to inspect the point by numerical simulations, because it reveals how the field enhancement is connected to the resonant behaviour. As can be seen from the results of Fig. 6, both silver (black solid squares) and gold (red solid triangles) field enhancement curves precisely resemble the ones of the resonant Q-factor.

11. T. Søndergaard and S. I. Bozhevolnyi, “Metal nano-strip optical resonators,” Opt. Express **15**, 4198–4204 (2007). [CrossRef] [PubMed]

*FWHM*of the corresponding resonance peak, even if the maximum seems to be slightly red-shifted by about 5 nm [see inset of Fig. 6(b), showing detuning effect on second order resonance for 300 nm wide silver and gold strips]. The 5nm red-shift of the maximum means that maximum field enhancement and maximum scattering are not found at the exact same wavelength. Anyway field enhancement at the edges results to be mostly of a resonant nature.

## 6. Estimation of the phase change due to reflection

*w*=20 nmand

*w*=30 nm) were also comprised in the analysis. Figure 7(a) shows the strip width as a function of the resonant wavelength for first, second and third order resonance (points). Numerically computed data have been fitted (according to the least-squares method) by linear equations:

*w*(

_{m}*λ*)=

*a*+

_{m}λ*b*;

_{m}*m*being the resonance order [see dashed lines in Fig. 7(a)]. Fitting parameters resulted to be:

*a*

_{1}=0.4786,

*a*

_{2}=1.0231,

*a*

_{3}=1.5446 and

*b*

_{1}=-155.57 nm,

*b*

_{2}=-305.45 nm,

*b*

_{3}=-436.88 nm. The squared correlation coefficient (R

^{2}) was greater than 0.999, thus ascertaining a fairly linear behaviour, which is certainly beneficial from a design point of view.

*λ*, the strip width fulfilling the

*m*-th order resonant condition is expected to be given by

*w*(

*λ*)=(

*mπ*-

*ϕ*)

*λ*/[2

*πn*(

_{eff}*λ*)]. As previously stated, the phase term

*ϕ*due to reflection at the edges is unknown, but its value can be estimated by fitting numerical data. In Fig. 7(a) we report such a prediction where we assume (wrongly) that the phase is independent of the wavelength (solid curves). Note significant deviations from exact data (points) and linear-fitting lines (dashed lines) too, even if the general trend is well reproduced. Furthermore, the estimated values for the phase resulted to be different for different orders, namely,

*ϕ*

_{1}=0.72 rad,

*ϕ*

_{2}=0.9 rad and

*ϕ*

_{3}=1.2 rad for first, second and third order resonance respectively. Though surprising, this behaviour is in agreement with what was reported in recent theory and experiment [13

13. T. Søndergaard, J. Beermann, A. Boltasseva, and S. I. Bozhevolnyi, “Slow-plasmon resonant nano-strip antennas: Analysis and demonstration,” Phys. Rev. B **77**, 115420-1–5 (2008). [CrossRef]

*ϕ*is expected to be strongly wavelength-dependent too. An estimation of this wavelength-dependent phase can be directly obtained from Eq. (2) according to the following equation:

*w*(

_{m}*λ*) the exact data (points) or the linear fitting curves (i.e. assuming for

*w*(

_{m}*λ*) the linear expressions given by numerical fitting of the exact data). Note that the phase (especially for higher order resonances) exhibits a strong dispersion in the short wavelength range, where also the effective index of SR-SPP is mostly non linear. Generally, the phase difference between the first and second order resonance results to be slightly larger than between second and third order resonances. Note also that higher order resonances exhibit a better agreement between phase estimation from exact data and phase estimation from linear fitting data.

## 7. Conclusion

13. T. Søndergaard, J. Beermann, A. Boltasseva, and S. I. Bozhevolnyi, “Slow-plasmon resonant nano-strip antennas: Analysis and demonstration,” Phys. Rev. B **77**, 115420-1–5 (2008). [CrossRef]

## Acknowledgments

## References and links

1. | H. Räther, |

2. | S. Lal, S. Link, and N. J. Halas, “Nano-optics from sensing to waveguiding,” Nat. Photonics |

3. | F. Wang and Y. R. Shen, “General properties of local plasmons in metal nanostructures,” Phys. Rev. Lett. |

4. | K. Imura, T. Nagahara, and H. Okamoto, “Near-field imaging of plasmon modes in gold nanorods,” J. Chem. Phys. |

5. | H. Ditlbacher, A. Hohenau, D. Wagner, U. Kreibig, M. Rogers, F. Hofer, F. R. Aussenegg, and J. R. Krenn, “Silver nanowires as surface plasmon resonators,” Phys. Rev. Lett. |

6. | T. Laroche and C. Girard, “Near-field optical properties of single plasmonic nanowires,” Appl. Phys. Lett. |

7. | F. Neubrech, et al., “Resonances of individual metal nanowires in the infrared,” Appl. Phys. Lett. |

8. | L. Novotny, “Effective wavelength scaling for optical antennas,” Phys. Rev. Lett. |

9. | J. Takahara, S. Yamagishi, H. Taki, A. Morimoto, and T. Kobayashi, “Guiding of a one-dimensional optical beam with nanometer diameter,” Opt. Lett. |

10. | T. Søndergaard and S. I. Bozhevolnyi, “Slow-plasmon resonant nanostructures: Scattering and field enhancements,” Phys. Rev. B |

11. | T. Søndergaard and S. I. Bozhevolnyi, “Metal nano-strip optical resonators,” Opt. Express |

12. | S. I. Bozhevolnyi and T. Søndergaard, “General properties of slow-plasmon resonant nanostructures: nanoantennas and resonators,” Opt. Express |

13. | T. Søndergaard, J. Beermann, A. Boltasseva, and S. I. Bozhevolnyi, “Slow-plasmon resonant nano-strip antennas: Analysis and demonstration,” Phys. Rev. B |

14. | T. Søndergaard, “Modeling of plasmonic nanostructures: Green’s function integral equation methods,” Phys. Status Solidi (b) |

15. | D. W. Prather, M. S. Mirotznik, and J. N. Mait, “Boundary integral methods applied to the analysis of diffractive optical elements,” J. Opt. Soc. Am. A |

16. | J. Jin, |

17. | E. N. Economou, “Surface plasmons in thin films,” Phys. Rev. |

18. | P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B |

19. | O. Svelto, |

**OCIS Codes**

(140.4780) Lasers and laser optics : Optical resonators

(240.6680) Optics at surfaces : Surface plasmons

(290.0290) Scattering : Scattering

**ToC Category:**

Optics at Surfaces

**History**

Original Manuscript: March 17, 2008

Revised Manuscript: April 21, 2008

Manuscript Accepted: April 21, 2008

Published: April 29, 2008

**Virtual Issues**

Vol. 3, Iss. 6 *Virtual Journal for Biomedical Optics*

**Citation**

G. Della Valle, T. Sondergaard, and S. I. Bozhevolnyi, "Plasmon-polariton nano-strip
resonators: from visible to infra-red," Opt. Express **16**, 6867-6876 (2008)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-16-10-6867

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### References

- H. Rather, Surface Plasmons (Springer, 1988).
- S. Lal, S. Link, and N. J. Halas, "Nano-optics from sensing to waveguiding," Nat. Photonics 1, 641-648 (2007). [CrossRef]
- F. Wang, and Y. R. Shen, "General properties of local plasmons in metal nanostructures," Phys. Rev. Lett. 97, 206806-1-4 (2006). [CrossRef] [PubMed]
- K. Imura, T. Nagahara, and H. Okamoto, "Near-field imaging of plasmon modes in gold nanorods," J. Chem. Phys. 122, 154701-1-5 (2005). [CrossRef] [PubMed]
- H. Ditlbacher, A. Hohenau, D. Wagner, U. Kreibig, M. Rogers, F. Hofer, F. R. Aussenegg, and J. R. Krenn, "Silver nanowires as surface plasmon resonators," Phys. Rev. Lett. 95, 257403-1-4 (2005). [CrossRef] [PubMed]
- T. Laroche and C. Girard, "Near-field optical properties of single plasmonic nanowires," Appl. Phys. Lett. 89, 233119-1-3 (2006). [CrossRef]
- F. Neubrech, et al., "Resonances of individual metal nanowires in the infrared," Appl. Phys. Lett. 89, 253104-1-3 (2006). [CrossRef]
- L. Novotny, "Effective wavelength scaling for optical antennas," Phys. Rev. Lett. 98, 266802-1-4 (2007). [CrossRef] [PubMed]
- J. Takahara, S. Yamagishi, H. Taki, A. Morimoto, and T. Kobayashi, "Guiding of a one-dimensional optical beam with nanometer diameter," Opt. Lett. 22, 475-477 (1997). [CrossRef] [PubMed]
- T. Søndergaard and S. I. Bozhevolnyi, "Slow-plasmon resonant nanostructures: Scattering and field enhancements," Phys. Rev. B 75, 073402-1-4 (2007). [CrossRef]
- T. Søndergaard and S. I. Bozhevolnyi, "Metal nano-strip optical resonators," Opt. Express 15, 4198-4204 (2007). [CrossRef] [PubMed]
- S. I. Bozhevolnyi and T. Søndergaard, "General properties of slow-plasmon resonant nanostructures: nanoantennas and resonators," Opt. Express 15, 10869-10877 (2007). [CrossRef] [PubMed]
- T. Søndergaard, J. Beermann, A. Boltasseva, and S. I. Bozhevolnyi, "Slow-plasmon resonant nano-strip antennas: Analysis and demonstration," Phys. Rev. B 77, 115420-1-5 (2008). [CrossRef]
- T. Søndergaard, "Modeling of plasmonic nanostructures: Green�??s function integral equation methods," Phys. Status Solidi(b) 244, 3448-3462 (2007). [CrossRef]
- D. W. Prather, M. S. Mirotznik, and J. N. Mait, "Boundary integral methods applied to the analysis of diffractive optical elements," J. Opt. Soc. Am. A 14, 34-43 (1997). [CrossRef]
- J. Jin, The Finite Element Method in Electromagnetics (John Wiley & Sons, New York 2002).
- E. N. Economou, "Surface plasmons in thin films," Phys. Rev. 182, 539-554 (1969). [CrossRef]
- P. B. Johnson and R. W. Christy, "Optical constants of the noble metals," Phys. Rev. B 6, 4370-4379 (1972). [CrossRef]
- O. Svelto, Principles of Lasers (Springer, 4th ed., 1998).

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