## Broadband converging plasmon resonance at a conical nanotip |

Optics Express, Vol. 21, Issue 5, pp. 6609-6617 (2013)

http://dx.doi.org/10.1364/OE.21.006609

Acrobat PDF (1127 KB)

### Abstract

We propose an analytical theory which predicts that Converging Plasmon Resonance (CPR) at conical nanotips exhibits a red-shifted and continuous band of resonant frequencies and suggests potential application of conical nanotips in various fields, such as plasmonic solar cells, photothermal therapy, tip-enhanced Raman and other spectroscopies. The CPR modes exhibit superior confinement and ten times broader scattering bandwidth over the entire solar spectrum than smooth nano-structures. The theory also explicitly connects the optimal angles and resonant optical frequencies to the material permittivities, with a specific optimum half angle that depends only on the real permittivity for high-permittivity and low-loss materials.

© 2013 OSA

## 1. Introduction

1. V. Ferry, J. Munday, and H. Atwater, “Design considerations for plasmonic photovoltaics,” Adv. Mater. **22**, 4794–4808 (2010) [CrossRef] [PubMed] .

2. X. Lu, M. Rycenga, S. Skrabalak, B. Wiley, and Y. Xia, “Chemical synthesis of novel plasmonic nanoparticles,” Annu. Rev. Phys. Chem. **60**, 167–192 (2009) [CrossRef] .

3. C. Lee, S. Bae, S. Mobasser, and H. Manohara, “A novel silicon nanotips antireflection surface for the micro sun sensor,” Nano Lett. **5**, 2438–2442 (2005) [CrossRef] [PubMed] .

5. A. V. Goncharenko, H. C. Chang, and J. K. Wang, “Electric near-field enhancing properties of a finite-size metal conical nano-tip,” Ultramicroscopy **107**, 151–157 (2007) [CrossRef] .

6. X. Huang, I. H. El-Sayed, W. Qian, and M. A. El-Sayed, “Cancer cell imaging and photothermal therapy in the near-infrared region by using gold nanorods,” J. Am. Chem. Soc **128**, 2115–2120 (2006) [CrossRef] [PubMed] .

7. R. Rodriguez-Oliveros and J. A. Sanchez-Gil, “Gold nanostars as thermoplasmonic nanoparticles for optical heating,” Opt. Express **20**, 621–626 (2012) [CrossRef] [PubMed] .

9. H. Shen, N. Guillot, J. Rouxel, M. de la Chapelle, and T. Toury, “Optimized plasmonic nanostructures for improved sensing activities,” Opt. Express **20**, 21278–21290 (2012) [CrossRef] [PubMed] .

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

12. A. Goncharenko, J. K. Wang, and Y. C. Chang, “Electric near-field enhancement of a sharp semi-infinite conical probe: Material and cone angle dependence,” Phys.Rev.B **74**, 235442 (2006) [CrossRef] .

## 2. Method description

### 2.1. Derivation of dispersion relationship

*α*and the cone is made of a material with a complex permittivity

*ε*and a permeability

_{m}*μ*surrounded by a dielectric medium with a real permittivity

_{m}*ε*and a permeability

_{o}*μ*, as depicted in the inset of Fig. 1(a). For the TM (transverse magnetic) mode with a magnetic field in the angular direction, the non-zero components of the EM fields are

_{o}*H*,

_{ϕ}*E*,

_{r}*E*, and the full set of Maxwell equations are simplified to a single scalar equation

_{θ}*ω*is the frequency of the incoming light, which is the same for the CPR resonance. The radially confined harmonics within the cone-like tips (0 <

*θ*<

*α*) are

*υ*is the eigenvalue),

*α*<

*θ*<

*π*) are

*θ*-dependence of the EM field suggests that the EM field does not decay away from the interface, in contrast to the logarithm decay in WKB theory [11

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

*υ*, with

12. A. Goncharenko, J. K. Wang, and Y. C. Chang, “Electric near-field enhancement of a sharp semi-infinite conical probe: Material and cone angle dependence,” Phys.Rev.B **74**, 235442 (2006) [CrossRef] .

*α*=

*π*/2 yields the classical planar plasmonic resonance condition

*ε*/

_{m}*ε*+ 1 = 0 and the frequency dependence of the complex permittivity

_{o}*ε*defines the planar plasmon resonant frequency

_{m}*ω*with this planar resonance condition [13

_{s}13. J. M. Pitarke, V. M. Silkin, E. V. Chulkov, and P. M. Echenique, “Theory of surface plasmons and surface-plasmon polaritons,” Rep. Prog. Phys **70**, 1–87 (2007) [CrossRef] .

*Re*[

*g*(

*α*,

*υ*)] grows rapidly as

*α*decreases below

*π*/2, TE resonance can never be realized [14

14. L. Novotny and S. J. Stranick, “Near-field optical microscopy and spectroscopy with pointed probes,” Annu. Rev. Phys. Chem. **57**, 303–331 (2006) [CrossRef] .

*E*scales as

_{r}*r*

^{υ−1}(r is the radial distance from the apex of the cone), while the azimuthal magnetic field

*H*is less singular since it scales as

_{ϕ}*r*(

^{υ}*υ*=

*υ*+

_{r}*iυ*). The restriction that the total EM energy be finite over the entire domain requires that

_{i}15. Y. Kawata, C. Xu, and W. Denk, “Feasibility of molecular-resolution fluorescence near-field microscopy using multi-photon absorption and field enhancement near a sharp tip,” J. Appl. Phys. **85**, 1294–1301 (1999) [CrossRef] .

*κ*=

*υ*/

_{i}*r*increases dramatically as CPR waves propagate toward the tip, which is found to be negative (

*υ*< 0) for the converging waves with speed retardation.

_{i}### 2.2. Conformal mapping of dispersion relationship

*g*(

*α*,

*υ*) from the complex

*υ*space to the complex permittivity

*ε*/

_{m}*ε*(

_{o}*ε*=

_{m}*ε*+

_{r}*iε*) space for cones (The dielectric permittivity

_{i}*ε*of the outside medium will be set to unity for simplicity). The CPR sustainable region (grey area in Fig. 1(a)) is confined within the bold curve (the image of

_{o}*υ*= 1) and the horizontal axis. The point

_{r}*κ*, is a branch point of the conformal map (

*dg*(

*α*,

*υ*)/

*dυ*= 0, marked as a solid circle in Fig. 1(a)). Another limit point (

*ε*= −1,

_{r}*ε*= 0,marked as an open circle in Fig. 1(a)) of the conformal map corresponds to the common large

_{i}*υ*limit for the pure dielectric (the image of the most singular mode with infinite imaginary part

_{r}*ε*<

_{ro}*ε*< −1). The gold permittivity data of classical Drude free electron theory [13

_{r}13. J. M. Pitarke, V. M. Silkin, E. V. Chulkov, and P. M. Echenique, “Theory of surface plasmons and surface-plasmon polaritons,” Rep. Prog. Phys **70**, 1–87 (2007) [CrossRef] .

*ε*∼ 0 and when maximum possible intensification occurs, CPR of Drude conical nanotips hence exhibits a continuous band of resonant frequencies, whose corresponding real permittivies vary from

_{i}*ε*to −1. This continuous broad band of resonant frequency is unique to conical nanotips due to their continuous change of length scale, while finite length scale of smooth nanoparticles only exhibits discrete and sharp resonance frequency. The intensification exponents 1 −

_{ro}*υ*computed from Eq. (1) for a Drude gold cone are shown as solid curves in Fig. 1(b) for different angles and wavelengths

_{r}*λ*of incident light. The entire solar spectrum is excited at small angles. In contrast, the extinction spectrum for a single nanosphere [2

2. X. Lu, M. Rycenga, S. Skrabalak, B. Wiley, and Y. Xia, “Chemical synthesis of novel plasmonic nanoparticles,” Annu. Rev. Phys. Chem. **60**, 167–192 (2009) [CrossRef] .

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

### 2.3. Global resonance frequency

*ε*/

_{i}*ε*of the material determines the energy loss-storage ratio of the system [17

_{r}17. F. Wang and Y. Shen, “General properties of local plasmons in metal nanostructures,” Phys. Rev. Lett. **97**, 206806 (2006) [CrossRef] [PubMed] .

*υ*computed from Eq. (1) for a gold cone with the permittivity model of [16

_{r}16. P. B. Johnson and R. W. Christy, “Optical constants of the nobel metals,” Phys. Rev. B **6**, 4370–4379 (1972) [CrossRef] .

*λ*of incident light. A global maximum in the intensification exponent exists at

*λ*= 892

*nm*and

*α*= 11°. As expected, this global resonant frequency resides at the minimum energy loss-storage ratio of gold, as shown in Fig. 2(b). The dash line connects local resonant frequency for different angles which ends at the global optimal angle. The resonance spectrum for a gold cone at different cone angles is shown in Fig. 2(a).

### 2.4. Asymptotic expansion

#### 2.4.1. Branch point

*g*(

*α*,

*υ*) in the limit of low conduction loss (

*ε*/

_{i}*ε*→ 0) is most relevant for both Drude-type and other metals at the

_{r}*IR*and visible frequency range. The five curves in the complex

*υ*plane shown in the inset of Fig. 3 are the loci of the dominant eigenvalues as a function of the half-angle

*α*for a cone with

*ε*= −182 and for five different imaginary permittivities

_{r}*ε*= 0.01, 0.5, 1, 3 and 10. Optimal angles for each

_{i}*ε*, marked as solid circles on the root loci, follows a horizontal line, which indicates that the optimal angles are only a function of

_{i}*ε*for metal with low conduction loss. This invariance to conduction loss is due to a curious asymptotic behavior of the dominant eigenvalues in the no-loss dielectric limit. Their smooth loci in Fig. 3 approach the real line on left half plane (

_{r}*υ*< 0,

_{r}*υ*= 0) and a vertical line (

_{i}*υ*< 0) at the pure dielectric limit of

_{i}*ε*→ 0, with the intersection of the two orthogonal lines occurring at

_{i}*α*=

*α*

^{*}. This intersection of the two lines at

*υ*= −1/2 and

*α*=

*α*

^{*}is the image of the branch point (

*ε*,

_{r}*ε*) = (

_{i}*ε*, 0). Given the wavelength of incident light and the material property, the branch point angle

_{ro}*α*=

*α*

^{*}can be obtained from

*dg*/

*dα*(

*α*

^{*}, −1/2) = 0. A leading order asymptotic estimate of the branch-point angle

*α*

^{*}for pure dielectric with |

*ε*| ≫ 1, leads to the following explicit relation

_{r}#### 2.4.2. Critical point

*dυ*/

_{r}*dα*= 0 simultaneously at the limit of

*ε*/

_{i}*ε*→ 0 and

_{r}*ε*/

_{r}*ε*→ −∞. Restricting ourselves to regions close to the critical point, an explicit Taylor expansion about the critical point can be carried out to yield

_{o}*υ*+ 1/2 = −

_{r}*iε*/

_{i}*g*(

_{υ}*α*,

_{o}*υ*). The optimal condition then reduces to

_{o}*ε*= 0) gives rise to the optimal angle

_{i}*α*and the corresponding

_{o}*υ*for any given

_{io}*ε*. Good agreement between the optimal angles predicted by Eq. (1) (solid curve in Fig. 3), critical local resonance points (dotted curve in Fig. 3) and numerical data from [18

_{r}18. N. A. Issa and R. Guckenberger, “Optical nanofocusing on tapered metallic waveguides,” Plasmonics **2**, 31–37 (2007) [CrossRef] .

*ε*between −200 and −2. Independent of

_{r}*ε*, both the branch point angles and critical point angles approach zero at the large

_{i}*ε*limit.

_{r}*ε*/

_{r}*ε*→ −∞,

_{o}*iε*/

_{r}*g*(

_{υ}*α*,

_{o}*υ*) approaches a constant and hence

_{o}*υ*+ 1/2 ∼ −0.693(

_{r}*ε*/

_{i}*ε*). This estimate of the dominant exponent at optimal angles for each frequency confirms that the global resonant frequency occurs at the minimum of the loss-storage ratio in Fig. 2(b). It also allows a convenient estimate of the local and global resonant frequencies from the material properties (

_{r}*ε*/

_{i}*ε*)(

_{r}*ω*). As seen in Fig. 3, both the predicted exponent and optimum angle at a particular real permittivity are in good agreement with numerical values from Eq. (1).

## 3. Comparison with numerical results

18. N. A. Issa and R. Guckenberger, “Optical nanofocusing on tapered metallic waveguides,” Plasmonics **2**, 31–37 (2007) [CrossRef] .

*E*| ∼

_{r}*r*

^{υr−1}, we obtain the dependence of the ratio of the electric field intensity on the exponents and the geometry

*E*/

_{tip}*E*= (

_{o}*r*

_{1}/

*r*

_{2})

^{υr−1}, where

*r*

_{1}is the radial distance from the apex of the cone to the point where

*E*is measured and

_{tip}*r*

_{2}is the radial distance from the apex to the point where laser with electric field

*E*impinges on the cone. From simple geometry,

_{o}*R*

_{1}=5 nm is the radius of curvature of the apex and

*R*

_{2}=300 nm is the radius of the cylindrical wire at the far end. Figure 4 favorably compares the literature data log

_{10}(

*E*/

_{tip}*E*) to our analytical estimate (

_{o}*υ*− 1)log

_{r}_{10}(

*R*

_{1}/

*R*

_{2}), where the exponents are evaluated from Eq. (1).

## 4. Conclusion

## Acknowledgments

## References and links

1. | V. Ferry, J. Munday, and H. Atwater, “Design considerations for plasmonic photovoltaics,” Adv. Mater. |

2. | X. Lu, M. Rycenga, S. Skrabalak, B. Wiley, and Y. Xia, “Chemical synthesis of novel plasmonic nanoparticles,” Annu. Rev. Phys. Chem. |

3. | C. Lee, S. Bae, S. Mobasser, and H. Manohara, “A novel silicon nanotips antireflection surface for the micro sun sensor,” Nano Lett. |

4. | A. R. Parker and H. E. Townley, “Biomimetics of photonic nanostructures,” Nat. Nanotechnol. |

5. | A. V. Goncharenko, H. C. Chang, and J. K. Wang, “Electric near-field enhancing properties of a finite-size metal conical nano-tip,” Ultramicroscopy |

6. | X. Huang, I. H. El-Sayed, W. Qian, and M. A. El-Sayed, “Cancer cell imaging and photothermal therapy in the near-infrared region by using gold nanorods,” J. Am. Chem. Soc |

7. | R. Rodriguez-Oliveros and J. A. Sanchez-Gil, “Gold nanostars as thermoplasmonic nanoparticles for optical heating,” Opt. Express |

8. | A. Mohammadi, F. Kaminski, V. Sandoghdar, and M. Agio, “Fluorescence enhancement with the optical (bi-) conical antenna,” J. Phys. Chem. C |

9. | H. Shen, N. Guillot, J. Rouxel, M. de la Chapelle, and T. Toury, “Optimized plasmonic nanostructures for improved sensing activities,” Opt. Express |

10. | A. F. Stevenson, “Solution of electromagnetic scattering problems as power series in the ratio (dimension of scatter)/wavelength,” Appl. Phys. Lett. |

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

12. | A. Goncharenko, J. K. Wang, and Y. C. Chang, “Electric near-field enhancement of a sharp semi-infinite conical probe: Material and cone angle dependence,” Phys.Rev.B |

13. | J. M. Pitarke, V. M. Silkin, E. V. Chulkov, and P. M. Echenique, “Theory of surface plasmons and surface-plasmon polaritons,” Rep. Prog. Phys |

14. | L. Novotny and S. J. Stranick, “Near-field optical microscopy and spectroscopy with pointed probes,” Annu. Rev. Phys. Chem. |

15. | Y. Kawata, C. Xu, and W. Denk, “Feasibility of molecular-resolution fluorescence near-field microscopy using multi-photon absorption and field enhancement near a sharp tip,” J. Appl. Phys. |

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

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

18. | N. A. Issa and R. Guckenberger, “Optical nanofocusing on tapered metallic waveguides,” Plasmonics |

**OCIS Codes**

(240.6680) Optics at surfaces : Surface plasmons

(350.5340) Other areas of optics : Photothermal effects

(350.6050) Other areas of optics : Solar energy

(310.6628) Thin films : Subwavelength structures, nanostructures

**ToC Category:**

Optics at Surfaces

**History**

Original Manuscript: January 3, 2013

Revised Manuscript: February 21, 2013

Manuscript Accepted: March 3, 2013

Published: March 8, 2013

**Citation**

Yunshan Wang, Franck Plouraboue, and Hsueh-Chia Chang, "Broadband converging plasmon resonance at a conical nanotip," Opt. Express **21**, 6609-6617 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-5-6609

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### References

- V. Ferry, J. Munday, and H. Atwater, “Design considerations for plasmonic photovoltaics,” Adv. Mater.22, 4794–4808 (2010). [CrossRef] [PubMed]
- X. Lu, M. Rycenga, S. Skrabalak, B. Wiley, and Y. Xia, “Chemical synthesis of novel plasmonic nanoparticles,” Annu. Rev. Phys. Chem.60, 167–192 (2009). [CrossRef]
- C. Lee, S. Bae, S. Mobasser, and H. Manohara, “A novel silicon nanotips antireflection surface for the micro sun sensor,” Nano Lett.5, 2438–2442 (2005). [CrossRef] [PubMed]
- A. R. Parker and H. E. Townley, “Biomimetics of photonic nanostructures,” Nat. Nanotechnol.2, 347–353 (2007). [CrossRef]
- A. V. Goncharenko, H. C. Chang, and J. K. Wang, “Electric near-field enhancing properties of a finite-size metal conical nano-tip,” Ultramicroscopy107, 151–157 (2007). [CrossRef]
- X. Huang, I. H. El-Sayed, W. Qian, and M. A. El-Sayed, “Cancer cell imaging and photothermal therapy in the near-infrared region by using gold nanorods,” J. Am. Chem. Soc128, 2115–2120 (2006). [CrossRef] [PubMed]
- R. Rodriguez-Oliveros and J. A. Sanchez-Gil, “Gold nanostars as thermoplasmonic nanoparticles for optical heating,” Opt. Express20, 621–626 (2012). [CrossRef] [PubMed]
- A. Mohammadi, F. Kaminski, V. Sandoghdar, and M. Agio, “Fluorescence enhancement with the optical (bi-) conical antenna,” J. Phys. Chem. C114, 7372–7377 (2010). [CrossRef]
- H. Shen, N. Guillot, J. Rouxel, M. de la Chapelle, and T. Toury, “Optimized plasmonic nanostructures for improved sensing activities,” Opt. Express20, 21278–21290 (2012). [CrossRef] [PubMed]
- A. F. Stevenson, “Solution of electromagnetic scattering problems as power series in the ratio (dimension of scatter)/wavelength,” Appl. Phys. Lett.24, 1134–1141 (1953).
- M. I. Stockman, “Nanofocusing of optical energy in tapered plasmonic waveguides,” Phys. Rev. Lett.93, 137404 (2004). [CrossRef] [PubMed]
- A. Goncharenko, J. K. Wang, and Y. C. Chang, “Electric near-field enhancement of a sharp semi-infinite conical probe: Material and cone angle dependence,” Phys.Rev.B74, 235442 (2006). [CrossRef]
- J. M. Pitarke, V. M. Silkin, E. V. Chulkov, and P. M. Echenique, “Theory of surface plasmons and surface-plasmon polaritons,” Rep. Prog. Phys70, 1–87 (2007). [CrossRef]
- L. Novotny and S. J. Stranick, “Near-field optical microscopy and spectroscopy with pointed probes,” Annu. Rev. Phys. Chem.57, 303–331 (2006). [CrossRef]
- Y. Kawata, C. Xu, and W. Denk, “Feasibility of molecular-resolution fluorescence near-field microscopy using multi-photon absorption and field enhancement near a sharp tip,” J. Appl. Phys.85, 1294–1301 (1999). [CrossRef]
- P. B. Johnson and R. W. Christy, “Optical constants of the nobel metals,” Phys. Rev. B6, 4370–4379 (1972). [CrossRef]
- F. Wang and Y. Shen, “General properties of local plasmons in metal nanostructures,” Phys. Rev. Lett.97, 206806 (2006). [CrossRef] [PubMed]
- N. A. Issa and R. Guckenberger, “Optical nanofocusing on tapered metallic waveguides,” Plasmonics2, 31–37 (2007). [CrossRef]

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