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

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2012 (2)

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]

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]

2011 (3)

P. Schattschneider and J. Verbeeck, “Theory of free electron vortices,” Ultramicroscopy 111, 1461–1468 (2011).
[CrossRef] [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,” Science 331, 192–195 (2011).
[CrossRef] [PubMed]

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]

2010 (5)

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

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

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

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

2009 (7)

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]

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]

W. Zhong, J. Xu, and X. Zhang, “Interaction of fast electron beam with photonic quasicrystals,” Opt. Express 17, 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. B 80, 113411 (2009).
[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]

A. Alu and N. Engheta, “The quest for magnetic plasmons at optical frequencies,” Opt. Express 17, 5723–5730 (2009).
[CrossRef] [PubMed]

2008 (2)

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]

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]

2007 (4)

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

A. Alu and N. Engheta, “Cloaking and transparency for collections of particles with metamaterial and plasmonic covers,” Opt. Express 15, 7578–7590 (2007).
[CrossRef] [PubMed]

V. M. Shalaev, “Optical negative-index metamaterials,” Nat. Photonics 1, 41–48 (2007).
[CrossRef]

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]

2006 (2)

2005 (3)

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 438, 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]

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]

2004 (2)

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]

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

2003 (2)

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 68, 245405 (2003).
[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,” Ultramicroscopy 96, 367–384 (2003).
[CrossRef] [PubMed]

2002 (1)

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]

1998 (1)

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]

1985 (1)

P. Batson, “Inelastic scattering of fast electrons in clusters of small spheres,” Surf. Sci. 156, 720–734 (1985).
[CrossRef]

1975 (1)

R. B. Pettit, J. Silcox, and R. Vincent, “Measurement of surface-plasmon dispersion in oxidized aluminum films,” Phys. Rev. B 11, 3116–3123 (1975).
[CrossRef]

1973 (1)

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

1972 (1)

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

1960 (1)

E. A. Stern and R. A. Ferrell, “Surface plasma oscillations of a degenerate electron gas,” Phys. Rev. 120, 130–136 (1960).
[CrossRef]

1958 (1)

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

1955 (1)

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]

1904 (1)

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).

Agarwal, A.

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 80, 113411 (2009).
[CrossRef]

Agrawal, A.

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]

Alu, A.

Anderson, I. M.

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]

Barfels, M. M. G.

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]

Batson, P.

P. Batson, “Inelastic scattering of fast electrons in clusters of small spheres,” Surf. Sci. 156, 720–734 (1985).
[CrossRef]

Bergman, J.

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]

Birkhoff, R. D.

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]

Blackstock, A. W.

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]

Bliokh, K. Y.

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).

Bliokh, Y. P.

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).

Boudarham, G.

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]

Brink, H. A.

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]

Burger, S.

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]

Burgner, R. P.

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]

Burokur, S. N.

Cai, W.

Carminati, R.

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 68, 245405 (2003).
[CrossRef]

Carozzi, T. D.

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]

Chen, C.

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]

Chettiar, U. K.

Christy, R. W.

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

Chu, M. W.

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]

de Lustrac, A.

Deng, J. P.

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]

Ditlbacher, H.

U. Hohenester, H. Ditlbacher, and J. R. Krenn, “Electron-energy-loss spectra of plasmonic nanoparticles,” Phys. Rev. Lett. 103, 106801 (2009).
[CrossRef] [PubMed]

Drachev, V. P.

Edwards, B. N.

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]

Egerton, R. F.

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

Engheta, N.

Enkrich, C.

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]

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]

Erni, R.

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 80, 113411 (2009).
[CrossRef]

Etrich, C.

Fernández-Domínguez, A. I.

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).
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Supplementary Material (2)

» Media 1: MPG (2170 KB)     
» Media 2: MPG (2286 KB)     

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Figures (6)

Fig. 1
Fig. 1

Schematic of U-shaped split ring resonator showing dimensions and axes.

Fig. 2
Fig. 2

Normal incidence transmission and reflection spectra for horizontal polarization.

Fig. 3
Fig. 3

Calculated vortex electron beam energy loss scattering probability of the SRR structure in Fig. 1 for 100 keV beam with transverse co-ordinate R0 = (0, 90 nm).

Fig. 4
Fig. 4

Map of vortex-EELS probability as a function of position for the 0.863 eV loss peak and a 100 keV beam. A video (Media 1) showing the vortex-EELS for different loss energies is provided online.

Fig. 5
Fig. 5

Map of EELS probability as a function of position for the 0.8 eV loss energy and a 100 keV beam. A video (Media 2) showing the EELS for different loss energies is provided online.

Fig. 6
Fig. 6

(a) Schematic of silver disk, (b) Electron energy loss scattering probability for a silver disk, for comparison with Fig. 2(e) of Ref. [33].

Equations (23)

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v = v θ y a x ^ + v θ x a y ^ + v z z ^ ,
j e ( r , t ) = e δ [ r r e ( t ) ] v ,
j m ( r , ω ) = i ω ε 0 × 1 ε r ( r , ω ) j e ( r , ω ) ,
( × v ) z ^ = 2 a v θ .
L z = m a v θ ,
L z = n h ¯ ,
v θ = n h ¯ m a .
e m δ [ r r e ( t ) ] = j m z v z = 2 e a ω ε v θ v z δ [ r r e ( t ) ] ,
e m = 2 e n h ¯ m ω ε v z a 2 .
Δ E = j m H ind d t = 0 h ¯ ω d ω Γ ( ω )
Δ E = e m v z H ind d t = 0 h ¯ ω d ω Γ ( ω ) .
H ind ( r , t ) = 1 2 π d ω e i ω t H ind ( r , ω ) ,
Γ ( ω ) = e m π h ¯ ω d t { e i ω t v z H ind [ r e ( t ) , ω ] }
H ( r , ω ) = i ω ε 0 d r G ¯ H ( r , r , ω ) j m ( r , ω ) .
Γ ( R 0 , ω ) = e m 2 v z 2 ε 0 π h ¯ d t d t { e i ω ( t t ) G z z H , ind [ r e ( t ) , r e ( t ) , ω ] } ,
× 1 ε r ( r , ω ) × G ¯ H ( r , r , ω ) ω 2 c 2 G ¯ H ( r , r , ω ) = δ ( r r ) I ¯ ,
Γ ( R 0 , ω ) = e m 2 ε 0 π h ¯ d z d z { e i ω ( z z ) / v z G z z H , ind [ ( R 0 , z ) , ( R 0 , z ) , ω ] } .
ω 2 c 2 G ¯ H ( r , r , ω ) = [ r × ] G ¯ E ( r , r , ω ) [ r × ] .
× × E ( r , ω ) k 0 2 ε r ( r , ω ) E ( r , ω ) = ε 0 k 0 2 P e ( r , ω ) = i ω μ 0 j e ( r , ω ) ,
× × G ¯ E ( r , ω ) k 0 2 ε ( r , ω ) G ¯ E ( r , ω ) = ε 0 k 0 2 δ ( r r ) ,
E ( r , ω ) = i ε 0 ω d r G ¯ E ( r , r , ω ) j e ( r , ω ) .
× 1 ε r ( r , ω ) × H ( r , ω ) k 0 2 H ( r , ω ) = × i ω ε r ( r , ω ) P e ( r , ω ) = × 1 ε r ( r , ω ) j e ( r , ω ) μ 0 k 0 2 P m ( r , ω ) i ω ε 0 j m ( r , ω ) ,
j m ( r , ω ) = i ω ε 0 × 1 ε r ( r , ω ) j e ( r , ω ) .

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