## The role of short and long range surface plasmons for plasmonic focusing applications

Optics Express, Vol. 17, Issue 16, pp. 14270-14280 (2009)

http://dx.doi.org/10.1364/OE.17.014270

Acrobat PDF (519 KB)

### Abstract

We propose and analyze a new plasmonic lens allowing the simultaneous focusing of both short and long range surface plasmons polaritons. The considered geometry is circularly symmetric and the SPP excitation is radially polarized. The long range and the short range modes are compared and found to have very different focusing properties. The trade-offs between the modes are discussed. The interplay between these two modes is used to demonstrate a practical focusing scenario providing a smaller spot size compared with previous version of plasmonic lenses, and a large depth of focus simultaneously.

© 2009 Optical Society of America

## 1. Introduction

1. Z. W. Liu, J. M. Steele, W. Srituravanich, Y. Pikus, C. Sun, and X. Zhang, “Focusing surface plasmons with a plasmonic lens,” Nano Lett. **5**, 1726–1729 (2005). [CrossRef] [PubMed]

2. W. Srituravanich, L. Pan, Y. Wang, C. Sun, C. Bogy, and X. Zhang, “Flying plasmonic lens in the near field for
high-speed nanolithography,” Nature Nanotech. **3**, 733–737 (2008). [CrossRef]

3. Q. Zhan, “Evanescent Bessel beam generation via surface plasmon resonance excitation by a radially polarized beam,” Opt. Lett. **31**, 1726–1728 (2006). [CrossRef] [PubMed]

4. A. Yanai and U. Levy, “Plasmonic focusing with a coaxial structure illuminated by radially polarized light,” Opt. Express **17**, 924–932 (2009). [CrossRef] [PubMed]

5. W. Chen and Q. Zhan, “Realization of an evanescent Bessel beam via surface plasmon interference excited by a radially polarized beam,” Opt. Lett. **34**, 722–724 (2009). [CrossRef] [PubMed]

6. G. Lerman, A. Yanai, and U. Levy, “Demonstration of nano focusing by the use of plasmonic lens illuminated with radially polarized light,” Nano Lett. **9**, 2139–2143 (2009). [CrossRef] [PubMed]

4. A. Yanai and U. Levy, “Plasmonic focusing with a coaxial structure illuminated by radially polarized light,” Opt. Express **17**, 924–932 (2009). [CrossRef] [PubMed]

*θ*.

## 2. SPP field distribution on a metallic circular disk

*β*) is directed along the radial direction. The structure is depicted in Fig. 1. The decay constants into the metal and the dielectric are designated by

*k*where i=M,D stands for the metallic and dielectric regions respectively.

_{i}*k*with

_{D}*k*with the appropriate sign resulting in:

_{M}*E*and

_{r}*ε*

_{i}*E*at z=0 one readily obtains that

_{z,i}*k*/

_{D}*k*= -

_{M}*ε*/

_{D}*ε*which is the same relation that holds for the standard 1D bi-layered plasmonic waveguide in Cartesian coordinates resulting in the well known characteristic equation

_{M}## 3. Focusing characteristics for long and short range SPPs for a circular disk

*E*) [8

_{r}8. D. Sarid, “Long-range surface-plasma waves on very thin metal films,” Phys. Rev. Lett. **47**, 1927–1930 (1981). [CrossRef]

9. P. Berini, “Plasmon-polariton waves guided by thin lossy metal films of finite width: bound modes of symmetric structures,” Phys. Rev. B **61**, 10484–10503 (2000). [CrossRef]

10. R. Charbonneau, N. Lahoud, G. Mattiussi, and P. Berini, “Demonstration of integrated optics elements based on long-ranging surface plasmon polaritons,” Opt. Express **13**, 977–984 (2005). [CrossRef] [PubMed]

11. A. Degiron and D. Smith, “Numerical simulations of long-range plasmons,” Opt. Express **14**, 1611–1625 (2006). [CrossRef] [PubMed]

12. K. Leosson, T. Nikolajsen, A. Boltasseva, and S. I. Bozhevolnyi, “Long-range surface plasmon polariton nanowire waveguides for device applications,” Opt. Express **14**, 314–319 (2006). [CrossRef] [PubMed]

9. P. Berini, “Plasmon-polariton waves guided by thin lossy metal films of finite width: bound modes of symmetric structures,” Phys. Rev. B **61**, 10484–10503 (2000). [CrossRef]

*ε*= -13.9 + 0.65i) embedded in a dielectric environment with refractive index n=1.33 (this environment will be later used in section 4). The excitation wavelength is

_{M}*λ*

_{0}=600 nm. Fig.2(a) and 2(b) show the results of the SRSPP mode for layer thickness of 15 and 10 nm respectively. A smaller central lobe is obtained for the 10 nm thick layer, indicating a shorter SPP wavelength. In addition, the contribution of the radial field becomes more significant as the layer thickness decreases (this will be further discussed in section 3.2). From Fig. 2(b), one can notice that at the edges of the PL the energy density decreases towards the middle of the structure. This is because of the very significant propagation losses for a 10 nm thick layer. A turning point is obtained around r=500 nm, where the focusing overcomes this loss. Finally, Fig.2(c) shows the result of a LRSPP mode for layer thickness of 15 nm. The

*E*component is seen to be negligible, indicating that the SPP wavelength of this mode is very similar to the vacuum wavelength. Further decrease of the layer has a negligible effect on the LRSPP mode.

_{r}### 3.1. Figures of merit

4. A. Yanai and U. Levy, “Plasmonic focusing with a coaxial structure illuminated by radially polarized light,” Opt. Express **17**, 924–932 (2009). [CrossRef] [PubMed]

*NA*

_{eff}=

*λ*

_{0}/(

*λ*×

_{SPP}*n*) (this criterion represents the improvement factor of the PL’s NA compared to that of a dielectric lens with a nearly zero focal length, i.e. having the highest possible NA for a given

_{D}*n*) and the propagation length

_{D}*L*= 1/(2×

*Im*[

*β*]). Fig. 3 presents these FOMs versus the excitation wavelength for a circular silver disk with metal thickness h=15 nm embedded in a dielectric medium. We vary the refractive index of the dielectric between

*n*= 1 and

_{D}*n*= 1.5. The results show the trade-off between the FOMs obtained by the LRSPP and the SRSPP modes. The SRSPP mode possesses a smaller SPP wavelength and therefore a smaller SPSZ at the cost of a smaller DOF due to tighter confinement to the interface. The LRSPP however, exhibits the opposite behavior. The effective NA provides an interesting comparison between the PL and a standard dielectric lens. Since the LRSPP evolves into the TEM mode of the surrounding dielectric as the metallic layer thickness decreases [9

_{D}9. P. Berini, “Plasmon-polariton waves guided by thin lossy metal films of finite width: bound modes of symmetric structures,” Phys. Rev. B **61**, 10484–10503 (2000). [CrossRef]

*NA*

_{eff}approaches the value of unity. For the SRSPP mode however,

*NA*

_{eff}grows significantly as the dielectric index or the optical frequency increase because of its highly dispersive nature. Unfortunately, the improvement in NA is coupled to higher loss. Nevertheless, an improvement factor of ≈ 1.5 is achievable with moderate loss.

### 3.2. Longitudinal and transverse electric components, and their effect on the spot size

**17**, 924–932 (2009). [CrossRef] [PubMed]

6. G. Lerman, A. Yanai, and U. Levy, “Demonstration of nano focusing by the use of plasmonic lens illuminated with radially polarized light,” Nano Lett. **9**, 2139–2143 (2009). [CrossRef] [PubMed]

*E*∣

_{z}^{2}/∣

*E*∣

_{r}^{2}= ∣

*k*∣

_{r}^{2}/∣

*k*∣

_{z}^{2}and ∣

*k*∣

_{r}^{2}> ∣

*k*∣

_{z}^{2}is a necessary condition to support a bound mode. Under radial polarization illumination the out of plane field interferes constructively at the center (see Eq. (11)) resulting in a smaller spot size compared to that obtained by linear polarization illumination [1

1. Z. W. Liu, J. M. Steele, W. Srituravanich, Y. Pikus, C. Sun, and X. Zhang, “Focusing surface plasmons with a plasmonic lens,” Nano Lett. **5**, 1726–1729 (2005). [CrossRef] [PubMed]

*E*) field broadens the spot size although it has no contribution to the total electric field density (∣

_{r}*E*∣

_{z}^{2}+∣

*E*∣

_{r}^{2}) at r=0. For a metallic layer thickness well above the metal skin depth the SPPs at each side are uncoupled and one can assume that the SPP dispersion relation is similar to the dispersion of an interface between semi-infinite metal and dielectric layers. For such a case ∣

*E*∣

_{z}^{2}/∣

*E*∣

_{r}^{2}= ∣

*k*∣2/∣

_{r}*k*∣

_{z}^{2}= ∣

*ε*∣/

_{M}*ε*> 1. However, as the metal thickness is decreased, the SRSPP evolves to reside more in the metal and the ratio ∣

_{D}*E*∣

_{z}^{2}/∣

*E*∣

_{r}^{2}= ∣

*k*∣

_{r}^{2}/∣

*k*∣

_{z}^{2}decreases towards unity. Therefore, we anticipate two opposite trends: while the ratio SPSZ/

*λ*

_{0}decreases with the decrease of the metal thickness (because

*λ*SPP decreases), the ratio SPSZ/

*λ*increases, because of the more significant contribution of the undesired

_{SPP}*E*component. In contrast, the LRSPP exhibits a nearly flat SPSZ/

_{r}*λ*ratio. This is because for this mode ∣

_{SPP}*E*∣ is negligible. These trends can be observed in Fig. 4, calculated for

_{r}*λ*0=600 nm.

## 4. Numerical analysis of a focusing scheme

*f*and thickness

*h*, surrounded by an outer wedge-like circular section with initial thickness

*w*. The overall radius of the structure is

*d*. The focusing is obtained at the center of the structure (

*r*=0). Wedge-like and adiabatic structures attract much interest in plasmonic research, primarily for confinement of electromagnetic waves [13

13. M. I. Stockman, “Nanofocusing of optical energy in tapered plasmonic waveguides,” Phys. Rev. Lett. **93**, 137404(2004). [CrossRef] [PubMed]

15. D. Gramotnev, M. Vogel, and M. Stockman, “Optimized nonadiabatic nanofocusing of plasmons by tapered metal rods,” J. Appl. Phys. **104**, 034311 (2008). [CrossRef]

16. E. Moreno, S. G. Rodrigo, S. I. Bozhevolnyi, L. Martin-Moreno, and F. J. Garcia-Vidal, “Guiding and focusing of electromagnetic fields with wedge plasmon-polaritons,” Phys. Rev. Lett. **100**, 023901-1-4 (2008). [CrossRef] [PubMed]

17. K. Kurihara, K. Yamamoto, J. Takahara, and A. Otomo,“Superfocusing modes of surface plasmon polaritons in a wedge-shaped geometry obtained by quasi-separation of variables,” J. Phys. A Math. Theor. **41**295401 (2008). [CrossRef]

18. E. Verhagen, A. Polman, and L. Kuipers, “Nanofocusing in laterally taperd plasmonic waveguides,” Opt. Express **16**, 45–57 (2008). [CrossRef] [PubMed]

19. L. Feng, D. Van Orden, M. Abashin, Q. Wang, Y. Chen, V. Lomakin, and Y. Fainman, “Nanoscale optical field localization by resonantly focused plasmons,” Opt. Express **17**, 4824–4832 (2009). [CrossRef] [PubMed]

20. A. Farjadpour, D. Roundy, A. Rodriguez, M. Ibanescu, P. Bermel, J. D. Joannopoulos, S. G. Johnson, and G. W. Burr, “Improving accuracy by subpixel smoothing in the finite-difference time domain,” Opt. Lett. **31**, 2972–2974 (2006). [CrossRef] [PubMed]

*λ*

_{0}= 600 nm. The geometrical parameters are

*w*=70 nm,

*d*=4200 nm,

*f*=2100 nm, and h=15 nm and computational cell resolution of 2 nm. Fig. 6 shows the resulting electric energy density patterns for both pure modes. It is apparent that in Fig. 6(a) that the energy density comprises of two contributions: 1 - contribution from the SRSPP mode that is bounded to the surface and 2 - contribution due to diffraction of light from the edges of the sample. Diffraction exists also in Fig. 6(b), but is less apparent because of the existence of the slowly decaying LRSPP mode.

*E*=

_{r}*J*

_{1}(

*αr*)×

*U*(

*r*-

*r*

_{0}), where

*α*is a scaling factor, r is the radial coordinate U(r) is the step function (U(r)=1 for r≥0, U(r)=0 for r<0) shifted by

*r*

_{0}. This incident field is an approximation to the donut shaped radially polarized polarized mode. This donut shape is favorable as it avoids light from incident at the thin metal section. Such light would partially penetrate through the thin metal resulting in the broadening of the focused spot.

*λ*

_{0}= 600 nm,

*α*= 0.1[

*μ*m

^{-1}], and

*r*

_{0}= 200 nm. The geometrical parameters are the same as those used for Fig. 6. The thin wedge is better observed in Fig. 7(b) which provides a zoom near the focal region. In a practical illumination scenario (as the one described above), the two modes (symmetric and anti-symmetric) are excited simultaneously. In order to evaluate the symmetry of the field at the focus, we define a function that measures the excited mode parity. The function compares the sign of the SPP magnetic component (

*H*

_{ϕ}) at two points on both sides of the metallic layer (z=±h/2, r=0) at each time step of the FDTD solver, and accumulates the sign comparisons over a single time period of the optical frequency. At each time step, in case the sign is equal, the function adds -1 to the accumulation. Otherwise, it adds +1. This can be formulated as:

*S*= (Δ

*t*/

*T*)∑

^{T/Δt}

_{n=1}

*Neq*[

*sgn*(

*H*

_{ϕ}(

*P*

_{1})),

*sgn*(

*H*

_{ϕ}(

*P*

_{2}))] where “S” is the symmetry measurement function, Δ

*t*is the time interval of the FDTD solver, T is the time period, “Neq[x,y]” is a function that returns -1 if x=y and +1 otherwise, and “sgn“ is the sign function. P

_{1}and P

_{2}are two measurement points at r=0 and z=±h/2. The normalization factor Δ

*t*/

*T*ensures that S returns a value in the range [-1,+1], where the cases S=-1 and S=+1 correspond to pure anti-symmetric and symmetric modes respectively.

## 5. Conclusions

*λ*

_{0}=600 nm (equivalent to a

*NA*

_{eff}of 1.61), and the LRSPP exhibits a slowly decaying field but with a larger spot size of 162 nm. Such a device may become useful for microscopy, nano-lithography, sensing and bio-sensing applications.

## Acknowledgments

## References and links

1. | Z. W. Liu, J. M. Steele, W. Srituravanich, Y. Pikus, C. Sun, and X. Zhang, “Focusing surface plasmons with a plasmonic lens,” Nano Lett. |

2. | W. Srituravanich, L. Pan, Y. Wang, C. Sun, C. Bogy, and X. Zhang, “Flying plasmonic lens in the near field for
high-speed nanolithography,” Nature Nanotech. |

3. | Q. Zhan, “Evanescent Bessel beam generation via surface plasmon resonance excitation by a radially polarized beam,” Opt. Lett. |

4. | A. Yanai and U. Levy, “Plasmonic focusing with a coaxial structure illuminated by radially polarized light,” Opt. Express |

5. | W. Chen and Q. Zhan, “Realization of an evanescent Bessel beam via surface plasmon interference excited by a radially polarized beam,” Opt. Lett. |

6. | G. Lerman, A. Yanai, and U. Levy, “Demonstration of nano focusing by the use of plasmonic lens illuminated with radially polarized light,” Nano Lett. |

7. | S. A. Maier, |

8. | D. Sarid, “Long-range surface-plasma waves on very thin metal films,” Phys. Rev. Lett. |

9. | P. Berini, “Plasmon-polariton waves guided by thin lossy metal films of finite width: bound modes of symmetric structures,” Phys. Rev. B |

10. | R. Charbonneau, N. Lahoud, G. Mattiussi, and P. Berini, “Demonstration of integrated optics elements based on long-ranging surface plasmon polaritons,” Opt. Express |

11. | A. Degiron and D. Smith, “Numerical simulations of long-range plasmons,” Opt. Express |

12. | K. Leosson, T. Nikolajsen, A. Boltasseva, and S. I. Bozhevolnyi, “Long-range surface plasmon polariton nanowire waveguides for device applications,” Opt. Express |

13. | M. I. Stockman, “Nanofocusing of optical energy in tapered plasmonic waveguides,” Phys. Rev. Lett. |

14. | D. K. Gramotnev and K. C. Vernon, “Adiabatic nano-focusing of plasmons by sharp metallic wedges,” Appl. Phys. B Lasers Opt. |

15. | D. Gramotnev, M. Vogel, and M. Stockman, “Optimized nonadiabatic nanofocusing of plasmons by tapered metal rods,” J. Appl. Phys. |

16. | E. Moreno, S. G. Rodrigo, S. I. Bozhevolnyi, L. Martin-Moreno, and F. J. Garcia-Vidal, “Guiding and focusing of electromagnetic fields with wedge plasmon-polaritons,” Phys. Rev. Lett. |

17. | K. Kurihara, K. Yamamoto, J. Takahara, and A. Otomo,“Superfocusing modes of surface plasmon polaritons in a wedge-shaped geometry obtained by quasi-separation of variables,” J. Phys. A Math. Theor. |

18. | E. Verhagen, A. Polman, and L. Kuipers, “Nanofocusing in laterally taperd plasmonic waveguides,” Opt. Express |

19. | L. Feng, D. Van Orden, M. Abashin, Q. Wang, Y. Chen, V. Lomakin, and Y. Fainman, “Nanoscale optical field localization by resonantly focused plasmons,” Opt. Express |

20. | A. Farjadpour, D. Roundy, A. Rodriguez, M. Ibanescu, P. Bermel, J. D. Joannopoulos, S. G. Johnson, and G. W. Burr, “Improving accuracy by subpixel smoothing in the finite-difference time domain,” Opt. Lett. |

21. |

**OCIS Codes**

(240.6680) Optics at surfaces : Surface plasmons

(310.2790) Thin films : Guided waves

(250.5403) Optoelectronics : Plasmonics

**ToC Category:**

Optoelectronics

**History**

Original Manuscript: May 21, 2009

Revised Manuscript: July 5, 2009

Manuscript Accepted: July 5, 2009

Published: July 31, 2009

**Citation**

Avner Yanai and Uriel Levy, "The role of short and long range surface plasmons for plasmonic focusing applications," Opt. Express **17**, 14270-14280 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-16-14270

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### References

- Z. W. Liu, J. M. Steele, W. Srituravanich, Y. Pikus, C. Sun, and X. Zhang, "Focusing surface plasmons with a plasmonic lens," Nano Lett. 5, 1726-1729 (2005). [CrossRef] [PubMed]
- W. Srituravanich, L. Pan, Y. Wang, C. Sun, C. Bogy, and X. Zhang, "Flying plasmonic lens in the near field for high-speed nanolithography," Nature Nanotech. 3, 733 - 737 (2008). [CrossRef]
- Q. Zhan, "Evanescent Bessel beam generation via surface plasmon resonance excitation by a radially polarized beam," Opt. Lett. 31, 1726-1728 (2006). [CrossRef] [PubMed]
- A. Yanai and U. Levy, "Plasmonic focusing with a coaxial structure illuminated by radially polarized light," Opt. Express 17, 924-932 (2009). [CrossRef] [PubMed]
- W. Chen and Q. Zhan, "Realization of an evanescent Bessel beam via surface plasmon interference excited by a radially polarized beam," Opt. Lett. 34, 722-724 (2009). [CrossRef] [PubMed]
- G. Lerman, A. Yanai and U. Levy, "Demonstration of nano focusing by the use of plasmonic lens illuminated with radially polarized light," Nano Lett. 9, 2139-2143 (2009). [CrossRef] [PubMed]
- S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, New York, 2007).
- D. Sarid, "Long-range surface-plasma waves on very thin metal films," Phys. Rev. Lett. 47, 1927-1930 (1981). [CrossRef]
- P. Berini, "Plasmon-polariton waves guided by thin lossy metal films of finite width: bound modes of symmetric structures," Phys. Rev. B 61, 10484-10503 (2000). [CrossRef]
- R. Charbonneau, N. Lahoud, G. Mattiussi, and P. Berini, "Demonstration of integrated optics elements based on long-ranging surface plasmon polaritons," Opt. Express 13, 977-984 (2005). [CrossRef] [PubMed]
- A. Degiron and D. Smith, "Numerical simulations of long-range plasmons," Opt. Express 14, 1611-1625 (2006). [CrossRef] [PubMed]
- K. Leosson, T. Nikolajsen, A. Boltasseva, and S. I. Bozhevolnyi, "Long-range surface plasmon polariton nanowire waveguides for device applications," Opt. Express 14, 314-319 (2006). [CrossRef] [PubMed]
- M. I. Stockman, "Nanofocusing of optical energy in tapered plasmonic waveguides," Phys. Rev. Lett. 93, 137404 (2004). [CrossRef] [PubMed]
- D. K. Gramotnev and K. C. Vernon, "Adiabatic nano-focusing of plasmons by sharp metallic wedges," Appl. Phys.B Lasers Opt. 86, 7-17 (2007).
- D. Gramotnev, M. Vogel, and M. Stockman, "Optimized nonadiabatic nanofocusing of plasmons by tapered metal rods," J. Appl. Phys. 104, 034311 (2008). [CrossRef]
- E. Moreno, S. G. Rodrigo, S. I. Bozhevolnyi, L. Martin-Moreno, and F. J. Garcia-Vidal, "Guiding and focusing of electromagnetic fields with wedge plasmon-polaritons," Phys. Rev. Lett. 100, 023901-1-4 (2008). [CrossRef] [PubMed]
- K. Kurihara, K. Yamamoto, J. Takahara and A. Otomo,"Superfocusing modes of surface plasmon polaritons in a wedge-shaped geometry obtained by quasi-separation of variables," J. Phys. A Math.Theor. 41295401 (2008). [CrossRef]
- E. Verhagen, A. Polman, and L. Kuipers, "Nanofocusing in laterally taperd plasmonic waveguides," Opt. Express 16, 45-57 (2008). [CrossRef] [PubMed]
- L. Feng, D. Van Orden, M. Abashin, Q. Wang, Y. Chen, V. Lomakin, and Y. Fainman, "Nanoscale optical field localization by resonantly focused plasmons," Opt. Express 17, 4824-4832 (2009). [CrossRef] [PubMed]
- A. Farjadpour, D. Roundy, A. Rodriguez, M. Ibanescu, P. Bermel, J. D. Joannopoulos, S. G. Johnson, and G. W. Burr, "Improving accuracy by subpixel smoothing in the finite-difference time domain," Opt. Lett. 31, 2972-2974 (2006). [CrossRef] [PubMed]
- http://ab-initio.mit.edu/meep/

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