Abstract

Following our recent development of a paradigm for extending the classic concepts of circuit elements to the infrared and optical frequencies [Phys. Rev. Lett. 95, 095504 (2005) ], in this paper we investigate the possibility of connecting nanoparticles in series and in parallel configurations, acting as nanocircuit elements. In particular, here we analyze a pair of conjoined half-cylinders whose relatively simple geometry may be studied and analyzed analytically. In this first part of this work, we derive a novel closed-form quasi-static analytical solution of the boundary-value problem associated with this geometry, which will be applied in Part 2 for a nanocircuit and physical interpretation of these results.

© 2007 Optical Society of America

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References

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  8. A. Alù and N. Engheta, "Theory of linear chains of metamaterial/plasmonic particles as sub-diffraction optical nanotransmission lines," Phys. Rev. B 74, 205436 (2006).
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  9. A. Alù, A. Salandrino, and N. Engheta, "Negative effective permeability and left-handed materials at optical frequencies," Opt. Express 14, 1557-1567 (2006).
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  14. A. N. Grigorenko, A. K. Geim, H. F. Gleeson, Y. Zhang, A. A. Firsov, I. Y. Khrushchev, and J. Petrovic, "Nanofabricated media with negative permeability at visible frequencies," Nature (London) 438, 335-338 (2005).
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  15. A. Alù and N. Engheta, "Three-dimensional nanotransmission lines at optical frequencies: a recipe for broadband negative-refraction optical metamaterials," Phys. Rev. B 75, 024304 (2007).
    [CrossRef]
  16. N. Engheta, A. Salandrino, and A. Alù, "Circuit elements at optical frequencies: nano-inductors, nano-capacitors, and nano-resistors," Phys. Rev. Lett. 95, 095504 (2005).
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  17. A. Alù, A. Salandrino, and N. Engheta, "Parallel, series, and intermediate interconnections of optical nanocircuit elements. 2. Nanocircuit and physical interpretation," J. Opt. Soc. Am. B 24, 3014-3022 (2007).
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    [CrossRef]

2007

A. Alù and N. Engheta, "Three-dimensional nanotransmission lines at optical frequencies: a recipe for broadband negative-refraction optical metamaterials," Phys. Rev. B 75, 024304 (2007).
[CrossRef]

A. Alù, A. Salandrino, and N. Engheta, "Parallel, series, and intermediate interconnections of optical nanocircuit elements. 2. Nanocircuit and physical interpretation," J. Opt. Soc. Am. B 24, 3014-3022 (2007).
[CrossRef]

2006

2005

A. N. Grigorenko, A. K. Geim, H. F. Gleeson, Y. Zhang, A. A. Firsov, I. Y. Khrushchev, and J. Petrovic, "Nanofabricated media with negative permeability at visible frequencies," Nature (London) 438, 335-338 (2005).
[CrossRef]

N. Engheta, A. Salandrino, and A. Alù, "Circuit elements at optical frequencies: nano-inductors, nano-capacitors, and nano-resistors," Phys. Rev. Lett. 95, 095504 (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]

2000

M. L. Brongersma, J. W. Hartman, and H. A. Atwater, "Electromagnetic energy transfer and switching in nanoparticle chain arrays below the diffraction limit," Phys. Rev. B 62, 16356 (2000).
[CrossRef]

1998

1993

R. W. Scharstein, "Mellin transform solution for the static line-source excitation of a dielectric wedge," IEEE Trans. Antennas Propag. 41, 1675-1679 (1993).
[CrossRef]

1991

Appl. Opt.

IEEE Trans. Antennas Propag.

R. W. Scharstein, "Mellin transform solution for the static line-source excitation of a dielectric wedge," IEEE Trans. Antennas Propag. 41, 1675-1679 (1993).
[CrossRef]

J. Opt. Soc. Am. B

Nature (London)

A. N. Grigorenko, A. K. Geim, H. F. Gleeson, Y. Zhang, A. A. Firsov, I. Y. Khrushchev, and J. Petrovic, "Nanofabricated media with negative permeability at visible frequencies," Nature (London) 438, 335-338 (2005).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. B

A. Alù and N. Engheta, "Three-dimensional nanotransmission lines at optical frequencies: a recipe for broadband negative-refraction optical metamaterials," Phys. Rev. B 75, 024304 (2007).
[CrossRef]

M. L. Brongersma, J. W. Hartman, and H. A. Atwater, "Electromagnetic energy transfer and switching in nanoparticle chain arrays below the diffraction limit," Phys. Rev. B 62, 16356 (2000).
[CrossRef]

A. Alù and N. Engheta, "Theory of linear chains of metamaterial/plasmonic particles as sub-diffraction optical nanotransmission lines," Phys. Rev. B 74, 205436 (2006).
[CrossRef]

Phys. Rev. Lett.

N. Engheta, A. Salandrino, and A. Alù, "Circuit elements at optical frequencies: nano-inductors, nano-capacitors, and nano-resistors," Phys. Rev. Lett. 95, 095504 (2005).
[CrossRef] [PubMed]

Science

G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, "Simultaneous negative phase and group velocity of light in a metamaterial," Science 312, 892-894 (2006).
[CrossRef] [PubMed]

Other

J. D. Jackson, Classical Electrodynamics (Wiley, 1999).

L. Landau and E. M. Lifschitz, Electrodynamics of Continuous Media (Pergamon, 1984).

M. Di Ventra, S. Evoy, and J. R. Heflin, Jr., Introduction to Nanoscale Science and Technology (Kluwer Academic, 2004).
[CrossRef]

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

C. Kittel, Introduction to Solid State Physics, 8th ed. (Wiley, 2005).

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Figures (5)

Fig. 1
Fig. 1

Geometry of the problem: two conjoined half-cylinders of different permittivities.

Fig. 2
Fig. 2

Schematic diagram of the mapping represented by Eq. (2). (a) The smallest circle, located inside the circle of inversion (dotted), is mapped into the circle above it. (b) Degeneration of the mapped circle in the case in which the original circumference passes through the center of the circle of inversion.

Fig. 3
Fig. 3

Mapping of the geometry of Fig. 1, after applying the Kelvin transformation [Eq. (2)] in the specific case that is illustrated in Fig. 2b, in order to obtain the double-wedge rectangular problem.

Fig. 4
Fig. 4

Transformed boundary-value problem after the Kelvin transform [Eq. (2)] and the translation [Eq. (10)]; a double wedge excited by an electric dipole forming an angle γ with the X axis equal to the angle formed by the uniform electric field with the normal to the interface between the two 2D half-cylinders in the original geometry (Fig. 1).

Fig. 5
Fig. 5

Potential distributions for a double wedge with permittivities ε 1 = ε 2 = 2 . The orientation of the exciting dipole is (a) γ = 0 , (b) γ = π 2 , and (c) γ = π 4 .

Equations (31)

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ρ 2 2 Φ ( ρ , φ ) ρ 2 + ρ Φ ( ρ , φ ) ρ + 2 Φ ( ρ , φ ) φ 2 = 0 .
ρ K = α 2 ρ ,
2 Φ ( ρ , φ ) = 0 2 Φ [ ρ K ( ρ ) , φ ] = 0 .
n ̂ = ρ ̂ sin φ ϕ ̂ cos φ ,
y ̂ = ρ ̂ sin φ + ϕ ̂ cos φ .
ϵ in Φ in [ ρ K ( ρ ) , φ ] y ̂ = ϵ 0 Φ 0 [ ρ K ( ρ ) , φ ] y ̂
ϵ in Φ in ( ρ , φ ) n ̂ = ϵ 0 Φ 0 ( ρ , φ ) n ̂
ϵ in [ Φ in [ ρ K ( ρ ) , φ ] ρ sin φ + 1 ρ Φ in [ ρ K ( ρ ) , φ ] φ cos φ ] = ϵ 0 [ Φ 0 [ ρ K ( ρ ) , φ ] ρ sin φ + 1 ρ Φ 0 [ ρ K ( ρ ) , φ ] φ cos φ ] .
ϵ in [ Φ in ( ρ , φ ) ρ sin φ + 1 ρ Φ in ( ρ , φ ) φ cos φ ] = ϵ 0 [ Φ 0 ( ρ , φ ) ρ sin φ + 1 ρ Φ 0 ( ρ , φ ) φ cos φ ] ,
Φ inc [ ρ K ( ρ ) , φ ] = 2 π ϵ 0 E 0 α 2 2 π ϵ 0 cos ( φ γ ) ρ K .
{ Y = y K α 2 2 R X = x K } , { ρ = X 2 + Y 2 φ = arctan ( Y X ) } .
f ̃ ( s ) = 0 ( ρ ) s 1 f ( ρ ) d ρ ,
f ( ρ ) = 1 2 π i c i c + i ( ρ ) s f ̃ ( s ) d s ,
Φ ̃ 0 + ( s , φ ) = A 0 + ( s ) cos ( s φ ) + B 0 + ( s ) sin ( s φ ) ,
Φ ̃ 0 ( s , φ ) = A 0 ( s ) cos ( s φ ) + B 0 ( s ) sin ( s φ ) ,
Φ ̃ 1 ( s , φ ) = A 1 ( s ) cos ( s φ ) + B 1 ( s ) sin ( s φ ) ,
Φ ̃ 2 ( s , φ ) = A 2 ( s ) cos ( s φ ) + B 2 ( s ) sin ( s φ ) .
Δ ( s ) = [ 4 ϵ 0 ϵ 1 ϵ 2 + ϵ 0 2 ( ϵ 1 + ϵ 2 ) + ϵ 1 ϵ 2 ( ϵ 1 + ϵ 2 ) + ( ϵ 0 + ϵ 1 ) ( ϵ 0 + ϵ 2 ) ( ϵ 1 + ϵ 2 ) cos ( π s ) ] sin 2 ( π s 2 ) = 0 .
A 0 + = A 1 = p r 0 s 1 sin ( γ ) 2 ϵ 0 sin ( s π 2 ) ,
B 0 + = p r 0 s 1 cos ( γ ) 2 ϵ 0 sin ( s π 2 ) ,
A 0 = p ϵ 0 r 0 s 1 cos ( γ ) cos ( s π 2 ) + p r 0 s 1 sin ( γ ) cos ( s π ) 2 ϵ 0 sin ( s π 2 ) ,
B 0 = p r 0 s 1 cos ( γ ) cos ( s π ) 2 ϵ 0 sin ( s π 2 ) p ϵ 0 r 0 s 1 sin ( γ ) cos ( s π 2 ) ,
B 1 = p r 0 s 1 cos ( γ ) 2 ϵ 1 sin ( s π 2 ) ,
A 2 = p r 0 s 1 cos ( γ ) cos ( s π 2 ) ϵ 1 + p r 0 s 1 sin ( γ ) cos ( π s ) 2 ϵ 0 sin ( s π 2 ) ,
B 2 = p r 0 s 1 cos ( γ ) cos ( s π ) 2 ϵ 1 sin ( s π 2 ) + p ϵ 0 r 0 s 1 sin ( γ ) cos ( s π 2 ) .
Φ ̃ i ( s , φ ) = r 0 s 1 p ϵ 0 ϵ i sin ( γ ) cos ( s φ ) ϵ 0 cos ( γ ) sin ( s φ ) 2 ϵ i sin ( s π 2 ) ,
Φ i ( ρ , φ ) = sgn ( ρ r 0 ) p N = 0 ϵ i sin ( γ ) cos ( 2 N φ ) ϵ 0 cos ( γ ) sin ( 2 N φ ) ϵ 0 ϵ i r 0 π cos ( N π ) × [ min ( r 0 , ρ ) max ( r 0 , ρ ) ] 2 n [ 1 1 2 δ ( N ) ] .
Φ i ( ρ , φ ) = p ( ρ 4 r 0 4 ) sin ( γ ) 2 π ϵ 0 r 0 [ ρ 4 + r 0 4 + 2 ρ 2 r 0 2 cos ( 2 φ ) ] + ϵ i p ρ 2 r 0 cos ( γ ) sin ( 2 φ ) ϵ 0 2 π [ ρ 4 + r 0 4 + 2 ρ 2 r 0 2 cos ( 2 φ ) ] ,
Φ i ( ρ , φ ) = Φ i parallel ( ρ , φ ) sin ( γ ) + Φ i series ( ρ , φ ) cos ( γ ) ,
{ Φ i parallel ( ρ , φ ) = E 0 [ 2 R ( ρ 2 + R 2 ) ρ 2 R cos ( 2 φ ) ρ ( ρ 2 + 4 R 2 ) sin ( φ ) ] ρ 2 + R 2 2 ρ R sin ( φ ) Φ i series ( ρ , φ ) = ϵ 0 E 0 ρ 2 cos ( φ ) [ 2 R sin ( φ ) ρ ] ϵ i [ ρ 2 + R 2 2 ρ R sin ( φ ) ] } ,
{ Φ i parallel ( ρ , φ ) = E 0 ρ 2 + R 2 ρ sin φ Φ i series ( ρ , φ ) = E 0 ϵ 0 ϵ i ρ 2 R 2 ρ cos φ } ,

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