## Vortex electron energy loss spectroscopy for near-field mapping of magnetic plasmons |

Optics Express, Vol. 20, Issue 14, pp. 15024-15034 (2012)

http://dx.doi.org/10.1364/OE.20.015024

Acrobat PDF (883 KB)

### Abstract

The theory of vortex electron beam electron energy loss spectroscopy (EELS), or vortex-EELS for short, is presented. This theory is applied, using Green function calculations within the finite-difference time-domain method, to calculate spatially resolved vortex-EELS maps of a metal split ring resonator (SRR). The vortex-EELS scattering cross section for the SRR structure is within an order of magnitude of conventional EELS typically for metal nanoparticles. This is promising in terms of feasibility for future measurements to map out the local magnetic response of metal nanostructures and to characterize their magnetic plasmon response in applications, including metamaterials.

© 2012 OSA

## 1. Introduction

2. W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photonics **1**, 224–227 (2007). [CrossRef]

4. B. Kanté, A. de Lustrac, J. M. Lourtioz, and S. N. Burokur, “Infrared cloaking based on the electric response of split ring resonators,” Opt. Express **16**, 9191–9198 (2008). [CrossRef] [PubMed]

5. D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science **305**, 788–792 (2004). [CrossRef] [PubMed]

6. V. M. Shalaev, W. Cai, U. K. Chettiar, H. K. Yuan, A. K. Sarychev, V. P. Drachev, and A. V. Kildishev, “Negative index of refraction in optical metamaterials,” Opt. Lett. **30**, 3356–3358 (2005). [CrossRef]

7. R. Merlin, “Metamaterials and the Landau-Lifshitz permeability argument: Large permittivity begets high-frequency magnetism,” Proc. Natl. Acad. Sci. U.S.A. **106**, 1693–1698 (2009). [CrossRef] [PubMed]

11. A. K. Sarychev, G. Shvets, and V. M. Shalaev, “Magnetic plasmon resonance,” Phys. Rev. E **73**, 036609 (2006). [CrossRef]

12. S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 terahertz,” Science **306**, 1351–1353 (2004). [CrossRef] [PubMed]

14. C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic metamaterials at telecommunication and visible frequencies,” Phys. Rev. Lett. **95**, 203901 (2005). [CrossRef] [PubMed]

15. A. W. Blackstock, R. H. Ritchie, and R. D. Birkhoff, “Mean free path for discrete electron energy losses in metallic foils,” Phys. Rev. **100**, 1078–1083 (1955). [CrossRef]

18. F. J. García de Abajo and A. Howie, “Relativistic electron energy loss and electron-induced photon emission in inhomogeneous dielectrics,” Phys. Rev. Lett. **80**, 5180–5183 (1998). [CrossRef]

19. R. Vincent and J. Silcox, “Dispersion of radiative surface plasmons in aluminum films by electron scattering,” Phys. Rev. Lett. **31**, 1487–1490 (1973). [CrossRef]

21. H. A. Brink, M. M. G. Barfels, R. P. Burgner, and B. N. Edwards, “A sub-50 meV spectrometer and energy filter for use in combination with 200 kv monochromated TEMs,” Ultramicroscopy **96**, 367–384 (2003). [CrossRef] [PubMed]

22. B. Schaffer, U. Hohenester, A. Trügler, and F. Hofer, “High-resolution surface plasmon imaging of gold nanoparticles by energy-filtered transmission electron microscopy,” Phys. Rev. B **79**, 041401 (2009). [CrossRef]

27. A. L. Koh, A. I. Fernández-Domínguez, D. W. McComb, S. A. Maier, and J. K. W. Yang, “High-resolution mapping of electron-beam-excited plasmon modes in lithographically defined gold nanostructures,” Nano Lett. **11**, 1323–1330 (2011). [CrossRef] [PubMed]

28. M. W. Chu, V. Myroshnychenko, C. Chen, J. P. Deng, C. Y. Mou, and F. J. García de Abajo, “Probing bright and dark surface-plasmon modes in individual and coupled nobel metal nanoparticles using an electron beam,” Nano Lett. **1**, 399–404 (2009). [CrossRef]

*magnetic*response of materials at the nanometer scale. Vortex electron beams have recently been demonstrated using diffractive phase-plates in transmission electron microscope setups [29

29. M. Uchida and A. Tonomura, “Generatio of electron beams carrying orbital angular momentum,” Nature **464**, 737–739 (2010). [CrossRef] [PubMed]

30. J. Verbeeck, H. Tian, and P. Schattschneider, “Production and application of electron vortex beams,” Nature **467**, 301–304 (2010). [CrossRef] [PubMed]

32. B. Thidé, H. Then, J. Sjöholm, K. Palmer, J. Bergman, T. D. Carozzi, Y. N. Istomin, N. H. Ibragimov, and R. Khamitova, “Utilization of photon orbital angular momentum in the low-frequency radio domain,” Phys. Rev. Lett. **99**, 087701 (2007). [CrossRef] [PubMed]

## 2. Vortex-EELS Theory

### 2.1. Effective Magnetic Charge

*a*, while moving with constant velocity in the

*z*-direction of

*v*. The total electron velocity is where

_{z}*x*̂ is the

*x*-directed unit vector. The electron current density is given by: where

*e*is the magnitude of the electron unit charge,

*r**(*

_{e}*t*) is the electron position as function of time and

*δ*is the Dirac delta function. The magnetic current density is given by where

*ω*is the angular frequency of the spiral.

*ε*

_{0}and

*ε**(*

_{r}**,**

*r**ω*) are the free-space and relative permittivities, respectively. Further description about the origin of this equation is given in the Appendix and the free space version is discussed elsewhere [34

34. P. E. Mayes, “The equivalence of electric and magnetic sources,” IEEE Trans. Antennas Propag. **6**, 295–296 (1958). [CrossRef]

*z*axis, so that To find

*v*, we use the angular momentum given by where

_{θ}*m*is the electron mass. In addition, we use the fact that the vortex beam has quantized orbital angular momentum given by where

*n*is the quantum number and

*h*̄ is the reduced Planck’s constant. Combining these two equations for the angular momentum gives We find the magnitude of the effective magnetic charge,

*e*, by writing the magnetic charge density along

_{m}*z*axis as follows: where

*ε*=

*ε*

_{0}

*ε*assuming

_{r}*ε*is scalar and

_{r}*j*=

_{mz}*j*(

_{mz}**,**

*r**t*) is the

*z*component of magnetic current. Subsequently, we obtain the effective magnetic charge:

### 2.2. Vortex-EELS Scattering Loss Probability

*E*, from a scattering object can be found from the Poynting theorem in the usual way [35], allowing for the inclusion of magnetic current: where

*H*^{ind}is the

*induced*magnetic field from current source and Γ(

*ω*) is the inelastic scattering probability. Here we use a semi-classical theory that considers a spiral electron particle motion with quantized angular momentum. We neglect energy loss that would typically result from the classical description of a spiraling electron beam. A quantum treatment of free electron vortices has been presented elsewhere [36

36. P. Schattschneider and J. Verbeeck, “Theory of free electron vortices,” Ultramicroscopy **111**, 1461–1468 (2011). [CrossRef] [PubMed]

37. F. J. García de Abajo, “Optical excitations in electron microscopy,” Rev. Mod. Phys. **82**, 209–275 (2010). [CrossRef]

*j**(*

_{m}**,**

*r**t*) into Eq. (10), we have The Fourier transform then gives where the property

*H*^{ind}(

**,**

*r**ω*) = [

*H*^{ind}(

**, −**

*r**ω*)]

^{*}has been used. From this, we find the inelastic scattering probability, through where ℑ indicates the imaginary part.

*Ḡ*^{H}, through Combining this with Eq. (13) and Eq. (8), we find the scattering probability for a given transverse coordinate

*R*_{0}= (

*x*

_{0},

*y*

_{0}): where

*R*_{0}is shown explicitly, and

*G*

^{H,ind}denotes the induced Green-tensor component obtained by subtracting the free-space Green green function. The Green function is defined in terms of the wave equation with a delta function excitation: where

**is the unit tensor. This wave equation is derived from the fully vectorial Maxwell equations, and the corresponding boundary conditions are those of Maxwell equations for an outgoing wave, for which the Green function goes to zero at infinity. In practice, we calculate the Green function using numerical computation of the comprehensive Maxwell equations with a magnetic dipole source in the next section.**

*Ī**z*(

_{e}*t*) =

*v*, the time integral of Eq. (15) can be expressed as an integral along magnetic charge trajectory, through Equation (17) is the desired result for our calculations. It remains to calculate the Green function component

_{z}t*G*for all positions along the trajectory of the magnetic charge produced by the vortex electron beam to calculate the vortex-EELS scattering probability. We will do this using finite-difference time-domain (FDTD) techniques, discussed below.

_{zz}## 3. Vortex-EELS for a Split Ring Resonator using FDTD

14. C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic metamaterials at telecommunication and visible frequencies,” Phys. Rev. Lett. **95**, 203901 (2005). [CrossRef] [PubMed]

### 3.1. FDTD Analysis of the SRR

*x*-direction and 330 nm along the

*y*-direction. The transmission and reflection was calculated using FDTD simulations with a normally-incident plane wave source of

*x*-polarization. The response of gold is taken from experiments [38

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

*x*- and

*y*-directions and perfectly matched layer boundaries in the positive and negative

*z*-direction. Figure 2 agrees quantitatively with past calculations using the finite element calculation method for the same structure [14

14. C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic metamaterials at telecommunication and visible frequencies,” Phys. Rev. Lett. **95**, 203901 (2005). [CrossRef] [PubMed]

### 3.2. Vortex-EELS Maps for the SRR

*R*_{0}= (0, 90 nm), as calculated using Eq. (17). The Green tensor component,

*z*-oriented magnetic dipole source in FDTD with perfectly matched layer boundary conditions, and monitoring the

*z*component of the magnetic field. Simulations were carried out for 30 positions of the magnetic dipole source positions with a 4 nm spacing. For dipoles within the lossy material, FDTD properly accounts for regularization of the Green tensor [39

39. C. P. Van Vlack and S. Hughes, “Finite-difference time domain technique as an efficient tool for obtaining the regularized green function: applications to the local field problem in quantum optics for inhomogeneous lossy materials,” Opt. Lett. (submitted) (2012). [PubMed]

30. J. Verbeeck, H. Tian, and P. Schattschneider, “Production and application of electron vortex beams,” Nature **467**, 301–304 (2010). [CrossRef] [PubMed]

40. B. J. McMorran, A. Agrawal, I. M. Anderson, A. A. Herzing, H. J. Lezec, J. J. McClelland, and J. Unguris, “Electron vortex beams with high quanta of orbital angular momentum,” Science **331**, 192–195 (2011). [CrossRef] [PubMed]

*n*= 1. The peak of the electron loss occurs at 0.863 eV, or 1437 nm, which is around the magnetic resonance in Fig. 2.

### 3.3. EELS Maps for the SRR

41. R. F. Egerton, *Electron Energy loss Spectroscopy in the Electron Microscope* (Springer, 2011). [CrossRef]

42. G. Boudarham, N. Feth, V. Myroshnychenko, S. Linden, F. J. García de Abajo, M. Wegener, and M. Kociak, “Spectral imaging of individual split-ring resonators,” Phys. Rev. Lett. **105**, 255501 (2010). [CrossRef]

43. C. Rockstuhl, F. Lederer, C. Etrich, S. Linden, T. Zentgraf, J. Kuhl, and H. Giessen, “On the reinterpretation of resonances in split-ring-resonators at normal incidence,” Opt. Express **14**, 8827–8836 (2006). [CrossRef] [PubMed]

## 4. Discussion

11. A. K. Sarychev, G. Shvets, and V. M. Shalaev, “Magnetic plasmon resonance,” Phys. Rev. E **73**, 036609 (2006). [CrossRef]

*n*= 1 and

*a*= 1 nm, which is consistent with recent experiments showing subnanometer focusing of vortex beams [45

45. P. Schattschneider, M. Stöger-Pollach, S. Löffler, A. Steiger-Thirsfeld, J. Hell, and J. Verbeeck, “Sub-nanometer free electrons with topological charge,” Ultramicroscopy **115**, 21–25 (2012). [CrossRef] [PubMed]

*n*(linear dependence) or decreasing the beam size

*a*(inverse square dependence). Since the magnetic charge is squared in the energy loss calculations, any improvements will be squared there as well. The increased

*n*has been demonstrated in experiment [40

40. B. J. McMorran, A. Agrawal, I. M. Anderson, A. A. Herzing, H. J. Lezec, J. J. McClelland, and J. Unguris, “Electron vortex beams with high quanta of orbital angular momentum,” Science **331**, 192–195 (2011). [CrossRef] [PubMed]

*a*as well.

## 5. Conclusions

## Appendix

*μ*= 1, one can show where

_{r}

*P**and*

_{e}*k*

_{0}are the electric polarization and wavenumber. Using an electric-dipole polarization source in the Maxwell equations, it is easy to derive the electric-field Green function equation: so that

**field: where we have introduced a magnetic-field polarization,**

*H*

*P**, and a magnetic-field current density. Equating coefficients then gives This gives Eq. (3). Note that the*

_{m}33. F. J. García de Abajo and M. Kociak, “Probing the photonic local density of states with electron energy loss spectroscopy,” Phys. Rev. Lett. **100**, 106804 (2008). [CrossRef] [PubMed]

33. F. J. García de Abajo and M. Kociak, “Probing the photonic local density of states with electron energy loss spectroscopy,” Phys. Rev. Lett. **100**, 106804 (2008). [CrossRef] [PubMed]

## Acknowledgments

## References and links

1. | L. D. Landau, L. P. Pitaevskii, and E. M. Lifshitz, |

2. | W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photonics |

3. | A. Alu and N. Engheta, “Cloaking and transparency for collections of particles with metamaterial and plasmonic covers,” Opt. Express |

4. | B. Kanté, A. de Lustrac, J. M. Lourtioz, and S. N. Burokur, “Infrared cloaking based on the electric response of split ring resonators,” Opt. Express |

5. | D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science |

6. | V. M. Shalaev, W. Cai, U. K. Chettiar, H. K. Yuan, A. K. Sarychev, V. P. Drachev, and A. V. Kildishev, “Negative index of refraction in optical metamaterials,” Opt. Lett. |

7. | R. Merlin, “Metamaterials and the Landau-Lifshitz permeability argument: Large permittivity begets high-frequency magnetism,” Proc. Natl. Acad. Sci. U.S.A. |

8. | V. M. Shalaev, “Optical negative-index metamaterials,” Nat. Photonics |

9. | R. Marqués, J. Martel, F. Mesa, and F. Medina, “Left-handed-media simulation and transmission of EM waves in subwavelength split-ring-resonator-loaded metallic waveguides,” Phys. Rev. Lett. |

10. | A. Alu and N. Engheta, “The quest for magnetic plasmons at optical frequencies,” Opt. Express |

11. | A. K. Sarychev, G. Shvets, and V. M. Shalaev, “Magnetic plasmon resonance,” Phys. Rev. E |

12. | S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 terahertz,” Science |

13. | A. N. Grigorenko, H. F. Gleeson, Y. Zhang, A. A. Firsov, I. Y. Khrushchev, and J. Petrovic, “Nanofabricated media with negative permeability at visible frequencies,” Nature |

14. | C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic metamaterials at telecommunication and visible frequencies,” Phys. Rev. Lett. |

15. | A. W. Blackstock, R. H. Ritchie, and R. D. Birkhoff, “Mean free path for discrete electron energy losses in metallic foils,” Phys. Rev. |

16. | E. A. Stern and R. A. Ferrell, “Surface plasma oscillations of a degenerate electron gas,” Phys. Rev. |

17. | P. Batson, “Inelastic scattering of fast electrons in clusters of small spheres,” Surf. Sci. |

18. | F. J. García de Abajo and A. Howie, “Relativistic electron energy loss and electron-induced photon emission in inhomogeneous dielectrics,” Phys. Rev. Lett. |

19. | R. Vincent and J. Silcox, “Dispersion of radiative surface plasmons in aluminum films by electron scattering,” Phys. Rev. Lett. |

20. | R. B. Pettit, J. Silcox, and R. Vincent, “Measurement of surface-plasmon dispersion in oxidized aluminum films,” Phys. Rev. B |

21. | H. A. Brink, M. M. G. Barfels, R. P. Burgner, and B. N. Edwards, “A sub-50 meV spectrometer and energy filter for use in combination with 200 kv monochromated TEMs,” Ultramicroscopy |

22. | B. Schaffer, U. Hohenester, A. Trügler, and F. Hofer, “High-resolution surface plasmon imaging of gold nanoparticles by energy-filtered transmission electron microscopy,” Phys. Rev. B |

23. | W. Zhong, J. Xu, and X. Zhang, “Interaction of fast electron beam with photonic quasicrystals,” Opt. Express |

24. | U. Hohenester, H. Ditlbacher, and J. R. Krenn, “Electron-energy-loss spectra of plasmonic nanoparticles,” Phys. Rev. Lett. |

25. | M. NǴom, S. Li, G. Schatz, R. Erni, A. Agarwal, N. Kotov, and T. B. Norris, “Electron-beam mapping of plasmon resonances in electromagnetically interacting gold nanorods,” Phys. Rev. B |

26. | F. J. García de Abajo, “Optical excitations in electron microscopy,” Rev. Mod. Phys. |

27. | A. L. Koh, A. I. Fernández-Domínguez, D. W. McComb, S. A. Maier, and J. K. W. Yang, “High-resolution mapping of electron-beam-excited plasmon modes in lithographically defined gold nanostructures,” Nano Lett. |

28. | M. W. Chu, V. Myroshnychenko, C. Chen, J. P. Deng, C. Y. Mou, and F. J. García de Abajo, “Probing bright and dark surface-plasmon modes in individual and coupled nobel metal nanoparticles using an electron beam,” Nano Lett. |

29. | M. Uchida and A. Tonomura, “Generatio of electron beams carrying orbital angular momentum,” Nature |

30. | J. Verbeeck, H. Tian, and P. Schattschneider, “Production and application of electron vortex beams,” Nature |

31. | K. Y. Bliokh, Y. P. Bliokh, S. Savelév, and F. Nori, “Semiclassical dynamics of electron wave packet states with phase vortices,” Phys. Rev. Lett. |

32. | B. Thidé, H. Then, J. Sjöholm, K. Palmer, J. Bergman, T. D. Carozzi, Y. N. Istomin, N. H. Ibragimov, and R. Khamitova, “Utilization of photon orbital angular momentum in the low-frequency radio domain,” Phys. Rev. Lett. |

33. | F. J. García de Abajo and M. Kociak, “Probing the photonic local density of states with electron energy loss spectroscopy,” Phys. Rev. Lett. |

34. | P. E. Mayes, “The equivalence of electric and magnetic sources,” IEEE Trans. Antennas Propag. |

35. | J. D. Jackson, |

36. | P. Schattschneider and J. Verbeeck, “Theory of free electron vortices,” Ultramicroscopy |

37. | F. J. García de Abajo, “Optical excitations in electron microscopy,” Rev. Mod. Phys. |

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

39. | C. P. Van Vlack and S. Hughes, “Finite-difference time domain technique as an efficient tool for obtaining the regularized green function: applications to the local field problem in quantum optics for inhomogeneous lossy materials,” Opt. Lett. (submitted) (2012). [PubMed] |

40. | B. J. McMorran, A. Agrawal, I. M. Anderson, A. A. Herzing, H. J. Lezec, J. J. McClelland, and J. Unguris, “Electron vortex beams with high quanta of orbital angular momentum,” Science |

41. | R. F. Egerton, |

42. | G. Boudarham, N. Feth, V. Myroshnychenko, S. Linden, F. J. García de Abajo, M. Wegener, and M. Kociak, “Spectral imaging of individual split-ring resonators,” Phys. Rev. Lett. |

43. | C. Rockstuhl, F. Lederer, C. Etrich, S. Linden, T. Zentgraf, J. Kuhl, and H. Giessen, “On the reinterpretation of resonances in split-ring-resonators at normal incidence,” Opt. Express |

44. | K. Joulain, R. Carminati, J. P. Mulet, and J. J. Greffet, “Definition and measurement of the local density of electromagnetic states close to an interface,” Phys. Rev. B |

45. | P. Schattschneider, M. Stöger-Pollach, S. Löffler, A. Steiger-Thirsfeld, J. Hell, and J. Verbeeck, “Sub-nanometer free electrons with topological charge,” Ultramicroscopy |

**OCIS Codes**

(240.6680) Optics at surfaces : Surface plasmons

(300.6330) Spectroscopy : Spectroscopy, inelastic scattering including Raman

(160.3918) Materials : Metamaterials

**ToC Category:**

Optics at Surfaces

**History**

Original Manuscript: April 13, 2012

Revised Manuscript: May 25, 2012

Manuscript Accepted: June 11, 2012

Published: June 20, 2012

**Citation**

Zeinab Mohammadi, Cole P. Van Vlack, Stephen Hughes, Jens Bornemann, and Reuven Gordon, "Vortex electron energy loss spectroscopy for near-field mapping of magnetic plasmons," Opt. Express **20**, 15024-15034 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-14-15024

Sort: Year | Journal | Reset

### References

- L. D. Landau, L. P. Pitaevskii, and E. M. Lifshitz, Electrodynamics of Continuous Media8, (Pergamon Press, 1984).
- W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photonics1, 224–227 (2007). [CrossRef]
- A. Alu and N. Engheta, “Cloaking and transparency for collections of particles with metamaterial and plasmonic covers,” Opt. Express15, 7578–7590 (2007). [CrossRef] [PubMed]
- B. Kanté, A. de Lustrac, J. M. Lourtioz, and S. N. Burokur, “Infrared cloaking based on the electric response of split ring resonators,” Opt. Express16, 9191–9198 (2008). [CrossRef] [PubMed]
- D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science305, 788–792 (2004). [CrossRef] [PubMed]
- V. M. Shalaev, W. Cai, U. K. Chettiar, H. K. Yuan, A. K. Sarychev, V. P. Drachev, and A. V. Kildishev, “Negative index of refraction in optical metamaterials,” Opt. Lett.30, 3356–3358 (2005). [CrossRef]
- R. Merlin, “Metamaterials and the Landau-Lifshitz permeability argument: Large permittivity begets high-frequency magnetism,” Proc. Natl. Acad. Sci. U.S.A.106, 1693–1698 (2009). [CrossRef] [PubMed]
- V. M. Shalaev, “Optical negative-index metamaterials,” Nat. Photonics1, 41–48 (2007). [CrossRef]
- R. Marqués, J. Martel, F. Mesa, and F. Medina, “Left-handed-media simulation and transmission of EM waves in subwavelength split-ring-resonator-loaded metallic waveguides,” Phys. Rev. Lett.89, 183901 (2002). [CrossRef] [PubMed]
- A. Alu and N. Engheta, “The quest for magnetic plasmons at optical frequencies,” Opt. Express17, 5723–5730 (2009). [CrossRef] [PubMed]
- A. K. Sarychev, G. Shvets, and V. M. Shalaev, “Magnetic plasmon resonance,” Phys. Rev. E73, 036609 (2006). [CrossRef]
- S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 terahertz,” Science306, 1351–1353 (2004). [CrossRef] [PubMed]
- A. N. Grigorenko, H. F. Gleeson, Y. Zhang, A. A. Firsov, I. Y. Khrushchev, and J. Petrovic, “Nanofabricated media with negative permeability at visible frequencies,” Nature438, 335 (2005). [CrossRef] [PubMed]
- C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic metamaterials at telecommunication and visible frequencies,” Phys. Rev. Lett.95, 203901 (2005). [CrossRef] [PubMed]
- A. W. Blackstock, R. H. Ritchie, and R. D. Birkhoff, “Mean free path for discrete electron energy losses in metallic foils,” Phys. Rev.100, 1078–1083 (1955). [CrossRef]
- E. A. Stern and R. A. Ferrell, “Surface plasma oscillations of a degenerate electron gas,” Phys. Rev.120, 130–136 (1960). [CrossRef]
- P. Batson, “Inelastic scattering of fast electrons in clusters of small spheres,” Surf. Sci.156, 720–734 (1985). [CrossRef]
- F. J. García de Abajo and A. Howie, “Relativistic electron energy loss and electron-induced photon emission in inhomogeneous dielectrics,” Phys. Rev. Lett.80, 5180–5183 (1998). [CrossRef]
- R. Vincent and J. Silcox, “Dispersion of radiative surface plasmons in aluminum films by electron scattering,” Phys. Rev. Lett.31, 1487–1490 (1973). [CrossRef]
- R. B. Pettit, J. Silcox, and R. Vincent, “Measurement of surface-plasmon dispersion in oxidized aluminum films,” Phys. Rev. B11, 3116–3123 (1975). [CrossRef]
- H. A. Brink, M. M. G. Barfels, R. P. Burgner, and B. N. Edwards, “A sub-50 meV spectrometer and energy filter for use in combination with 200 kv monochromated TEMs,” Ultramicroscopy96, 367–384 (2003). [CrossRef] [PubMed]
- B. Schaffer, U. Hohenester, A. Trügler, and F. Hofer, “High-resolution surface plasmon imaging of gold nanoparticles by energy-filtered transmission electron microscopy,” Phys. Rev. B79, 041401 (2009). [CrossRef]
- W. Zhong, J. Xu, and X. Zhang, “Interaction of fast electron beam with photonic quasicrystals,” Opt. Express17, 13270–13282 (2009). [CrossRef] [PubMed]
- U. Hohenester, H. Ditlbacher, and J. R. Krenn, “Electron-energy-loss spectra of plasmonic nanoparticles,” Phys. Rev. Lett.103, 106801 (2009). [CrossRef] [PubMed]
- M. NǴom, S. Li, G. Schatz, R. Erni, A. Agarwal, N. Kotov, and T. B. Norris, “Electron-beam mapping of plasmon resonances in electromagnetically interacting gold nanorods,” Phys. Rev. B80, 113411 (2009). [CrossRef]
- F. J. García de Abajo, “Optical excitations in electron microscopy,” Rev. Mod. Phys.82, 209–275 (2010). [CrossRef]
- A. L. Koh, A. I. Fernández-Domínguez, D. W. McComb, S. A. Maier, and J. K. W. Yang, “High-resolution mapping of electron-beam-excited plasmon modes in lithographically defined gold nanostructures,” Nano Lett.11, 1323–1330 (2011). [CrossRef] [PubMed]
- M. W. Chu, V. Myroshnychenko, C. Chen, J. P. Deng, C. Y. Mou, and F. J. García de Abajo, “Probing bright and dark surface-plasmon modes in individual and coupled nobel metal nanoparticles using an electron beam,” Nano Lett.1, 399–404 (2009). [CrossRef]
- M. Uchida and A. Tonomura, “Generatio of electron beams carrying orbital angular momentum,” Nature464, 737–739 (2010). [CrossRef] [PubMed]
- J. Verbeeck, H. Tian, and P. Schattschneider, “Production and application of electron vortex beams,” Nature467, 301–304 (2010). [CrossRef] [PubMed]
- K. Y. Bliokh, Y. P. Bliokh, S. Savelév, and F. Nori, “Semiclassical dynamics of electron wave packet states with phase vortices,” Phys. Rev. Lett.99, 190404 (2007).
- B. Thidé, H. Then, J. Sjöholm, K. Palmer, J. Bergman, T. D. Carozzi, Y. N. Istomin, N. H. Ibragimov, and R. Khamitova, “Utilization of photon orbital angular momentum in the low-frequency radio domain,” Phys. Rev. Lett.99, 087701 (2007). [CrossRef] [PubMed]
- F. J. García de Abajo and M. Kociak, “Probing the photonic local density of states with electron energy loss spectroscopy,” Phys. Rev. Lett.100, 106804 (2008). [CrossRef] [PubMed]
- P. E. Mayes, “The equivalence of electric and magnetic sources,” IEEE Trans. Antennas Propag.6, 295–296 (1958). [CrossRef]
- J. D. Jackson, Classical Electrodynamics (Wiley, 1999).
- P. Schattschneider and J. Verbeeck, “Theory of free electron vortices,” Ultramicroscopy111, 1461–1468 (2011). [CrossRef] [PubMed]
- F. J. García de Abajo, “Optical excitations in electron microscopy,” Rev. Mod. Phys.82, 209–275 (2010). [CrossRef]
- P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B6, 4370–4379 (1972). [CrossRef]
- C. P. Van Vlack and S. Hughes, “Finite-difference time domain technique as an efficient tool for obtaining the regularized green function: applications to the local field problem in quantum optics for inhomogeneous lossy materials,” Opt. Lett. (submitted) (2012). [PubMed]
- B. J. McMorran, A. Agrawal, I. M. Anderson, A. A. Herzing, H. J. Lezec, J. J. McClelland, and J. Unguris, “Electron vortex beams with high quanta of orbital angular momentum,” Science331, 192–195 (2011). [CrossRef] [PubMed]
- R. F. Egerton, Electron Energy loss Spectroscopy in the Electron Microscope (Springer, 2011). [CrossRef]
- G. Boudarham, N. Feth, V. Myroshnychenko, S. Linden, F. J. García de Abajo, M. Wegener, and M. Kociak, “Spectral imaging of individual split-ring resonators,” Phys. Rev. Lett.105, 255501 (2010). [CrossRef]
- C. Rockstuhl, F. Lederer, C. Etrich, S. Linden, T. Zentgraf, J. Kuhl, and H. Giessen, “On the reinterpretation of resonances in split-ring-resonators at normal incidence,” Opt. Express14, 8827–8836 (2006). [CrossRef] [PubMed]
- K. Joulain, R. Carminati, J. P. Mulet, and J. J. Greffet, “Definition and measurement of the local density of electromagnetic states close to an interface,” Phys. Rev. B68, 245405 (2003). [CrossRef]
- P. Schattschneider, M. Stöger-Pollach, S. Löffler, A. Steiger-Thirsfeld, J. Hell, and J. Verbeeck, “Sub-nanometer free electrons with topological charge,” Ultramicroscopy115, 21–25 (2012). [CrossRef] [PubMed]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.