Abstract

We present here an analytical quasi-static circuit model for the coupling among small nanoparticles excited by an optical electric field in the framework of the optical lumped nanocircuit theory [N. Engheta, A. Salandrino, and A. Alù, Phys. Rev. Lett. 95, 095504 (2005)]. We derive how coupling effects may affect the corresponding nanocircuit model by adding lumped controlled sources that depend on the optical voltages applied on the coupled particles as coupled lumped elements. With the same technique, we may model the presence of a substrate located underneath the nanocircuit elements, relating its presence to the coupling with a properly modeled image nanoparticle. These results are of importance in the understanding and the design of complex optical nanocircuits at infrared and optical frequencies.

© 2007 Optical Society of America

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  1. 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]
  2. A. Alù and N. Engheta, "Optical nano-transmission lines: synthesis of planar left-handed metamaterials in the infrared and visible regimes," J. Opt. Soc. Am. B 23, 571-583 (2006).
    [CrossRef]
  3. 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]
  4. 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]
  5. A. Salandrino, A. Alù, and N. Engheta, "Parallel, series, and intermediate interconnections of optical nanocircuit elements - Part 1: Analytical Solution," submitted to J. Opt. Soc. Am. B. (Manuscript can be viewed at http://arxiv.org/abs/0707.1002>)
  6. A. Alù, A. Salandrino, and N. Engheta, "Parallel, series, and intermediate interconnections of optical nanocircuit elements - Part 2: Nanocircuit and Physical Interpretation," submitted to J. Opt. Soc. Am. B. (Manuscript can be viewed at http://arxiv.org/abs/0707.1003>)
  7. R. W. Rendell, and D. J. Scalapino, "Surface plasmons confined by microstructures on tunnel junctions," Phys. Rev. B 24, 3276-3294 (1981).
    [CrossRef]
  8. X. C. Zeng, P. M. Hui, D. J. Bergman, and D. Stroud, "Correlation and clustering in the optical properties of composites: a numerical study," Phys. Rev. B 39, 13224-13230 (1989).
    [CrossRef]
  9. D. A. Genov, A. K. Sarychev, V. M. Shalaev, and A. Wei, "Resonant field enhancement from metal nanoparticle arrays," Nano Lett. 4, 153-158 (2004).
    [CrossRef]
  10. A. I. Csurgay, and W. Porod, "Surface plasmon waves in nanoelectronic circuits," Int. J. Circuit Theory and Applications 32, 339-361 (2004).Q1
    [CrossRef]
  11. P. M. Morse and H. Feshbach, Methods of Theoretical Physics, Part I (McGraw-Hill, New York, 1953), p. 663.
  12. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
  13. J. Gómez Rivas, C. Janke, P. Bolivar, and H. Kurz, "Transmission of THz radiation through InSb gratings of subwavelength apertures," Opt. Express 13, 847-859 (2005).
    [CrossRef] [PubMed]
  14. J. D. Jackson, Classical Electrodynamics (Wiley, New York, USA, 1999).
  15. A. Alù, M. G. Silveirinha, A. Salandrino, and N. Engheta, "Epsilon-near-zero metamaterials and electromagnetic sources: tailoring the radiation phase pattern," Phys. Rev. B 75, 155410 (2007).
    [CrossRef]
  16. M. G. Silveirinha and N. Engheta, "Tunneling of electromagnetic energy through subwavelength channels and bends using ?-near-zero materials," Phys. Rev. Lett. 97, 157403 (2006).
    [CrossRef] [PubMed]
  17. M. G. Silveirinha, A. Alù, J. Li, and N. Engheta, "Nanoinsulators and nanoconnectors for optical nanocircuits," submitted for publication. (Manuscript can be viewed at http://arxiv.org/abs/cond-mat/0703600>).

2007 (2)

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, M. G. Silveirinha, A. Salandrino, and N. Engheta, "Epsilon-near-zero metamaterials and electromagnetic sources: tailoring the radiation phase pattern," Phys. Rev. B 75, 155410 (2007).
[CrossRef]

2006 (3)

M. G. Silveirinha and N. Engheta, "Tunneling of electromagnetic energy through subwavelength channels and bends using ?-near-zero materials," Phys. Rev. Lett. 97, 157403 (2006).
[CrossRef] [PubMed]

A. Alù and N. Engheta, "Optical nano-transmission lines: synthesis of planar left-handed metamaterials in the infrared and visible regimes," J. Opt. Soc. Am. B 23, 571-583 (2006).
[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]

2005 (2)

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]

J. Gómez Rivas, C. Janke, P. Bolivar, and H. Kurz, "Transmission of THz radiation through InSb gratings of subwavelength apertures," Opt. Express 13, 847-859 (2005).
[CrossRef] [PubMed]

2004 (2)

D. A. Genov, A. K. Sarychev, V. M. Shalaev, and A. Wei, "Resonant field enhancement from metal nanoparticle arrays," Nano Lett. 4, 153-158 (2004).
[CrossRef]

A. I. Csurgay, and W. Porod, "Surface plasmon waves in nanoelectronic circuits," Int. J. Circuit Theory and Applications 32, 339-361 (2004).Q1
[CrossRef]

1989 (1)

X. C. Zeng, P. M. Hui, D. J. Bergman, and D. Stroud, "Correlation and clustering in the optical properties of composites: a numerical study," Phys. Rev. B 39, 13224-13230 (1989).
[CrossRef]

1981 (1)

R. W. Rendell, and D. J. Scalapino, "Surface plasmons confined by microstructures on tunnel junctions," Phys. Rev. B 24, 3276-3294 (1981).
[CrossRef]

Alù, A.

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, M. G. Silveirinha, A. Salandrino, and N. Engheta, "Epsilon-near-zero metamaterials and electromagnetic sources: tailoring the radiation phase pattern," Phys. Rev. B 75, 155410 (2007).
[CrossRef]

A. Alù and N. Engheta, "Optical nano-transmission lines: synthesis of planar left-handed metamaterials in the infrared and visible regimes," J. Opt. Soc. Am. B 23, 571-583 (2006).
[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]

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]

Bergman, D. J.

X. C. Zeng, P. M. Hui, D. J. Bergman, and D. Stroud, "Correlation and clustering in the optical properties of composites: a numerical study," Phys. Rev. B 39, 13224-13230 (1989).
[CrossRef]

Bolivar, P.

Csurgay, A. I.

A. I. Csurgay, and W. Porod, "Surface plasmon waves in nanoelectronic circuits," Int. J. Circuit Theory and Applications 32, 339-361 (2004).Q1
[CrossRef]

Engheta, N.

A. Alù, M. G. Silveirinha, A. Salandrino, and N. Engheta, "Epsilon-near-zero metamaterials and electromagnetic sources: tailoring the radiation phase pattern," Phys. Rev. B 75, 155410 (2007).
[CrossRef]

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 and N. Engheta, "Optical nano-transmission lines: synthesis of planar left-handed metamaterials in the infrared and visible regimes," J. Opt. Soc. Am. B 23, 571-583 (2006).
[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]

M. G. Silveirinha and N. Engheta, "Tunneling of electromagnetic energy through subwavelength channels and bends using ?-near-zero materials," Phys. Rev. Lett. 97, 157403 (2006).
[CrossRef] [PubMed]

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]

Genov, D. A.

D. A. Genov, A. K. Sarychev, V. M. Shalaev, and A. Wei, "Resonant field enhancement from metal nanoparticle arrays," Nano Lett. 4, 153-158 (2004).
[CrossRef]

Gómez Rivas, J.

Hui, P. M.

X. C. Zeng, P. M. Hui, D. J. Bergman, and D. Stroud, "Correlation and clustering in the optical properties of composites: a numerical study," Phys. Rev. B 39, 13224-13230 (1989).
[CrossRef]

Janke, C.

Kurz, H.

Porod, W.

A. I. Csurgay, and W. Porod, "Surface plasmon waves in nanoelectronic circuits," Int. J. Circuit Theory and Applications 32, 339-361 (2004).Q1
[CrossRef]

Rendell, R. W.

R. W. Rendell, and D. J. Scalapino, "Surface plasmons confined by microstructures on tunnel junctions," Phys. Rev. B 24, 3276-3294 (1981).
[CrossRef]

Salandrino, A.

A. Alù, M. G. Silveirinha, A. Salandrino, and N. Engheta, "Epsilon-near-zero metamaterials and electromagnetic sources: tailoring the radiation phase pattern," Phys. Rev. B 75, 155410 (2007).
[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]

Sarychev, A. K.

D. A. Genov, A. K. Sarychev, V. M. Shalaev, and A. Wei, "Resonant field enhancement from metal nanoparticle arrays," Nano Lett. 4, 153-158 (2004).
[CrossRef]

Scalapino, D. J.

R. W. Rendell, and D. J. Scalapino, "Surface plasmons confined by microstructures on tunnel junctions," Phys. Rev. B 24, 3276-3294 (1981).
[CrossRef]

Shalaev, V. M.

D. A. Genov, A. K. Sarychev, V. M. Shalaev, and A. Wei, "Resonant field enhancement from metal nanoparticle arrays," Nano Lett. 4, 153-158 (2004).
[CrossRef]

Silveirinha, M. G.

A. Alù, M. G. Silveirinha, A. Salandrino, and N. Engheta, "Epsilon-near-zero metamaterials and electromagnetic sources: tailoring the radiation phase pattern," Phys. Rev. B 75, 155410 (2007).
[CrossRef]

M. G. Silveirinha and N. Engheta, "Tunneling of electromagnetic energy through subwavelength channels and bends using ?-near-zero materials," Phys. Rev. Lett. 97, 157403 (2006).
[CrossRef] [PubMed]

Stroud, D.

X. C. Zeng, P. M. Hui, D. J. Bergman, and D. Stroud, "Correlation and clustering in the optical properties of composites: a numerical study," Phys. Rev. B 39, 13224-13230 (1989).
[CrossRef]

Wei, A.

D. A. Genov, A. K. Sarychev, V. M. Shalaev, and A. Wei, "Resonant field enhancement from metal nanoparticle arrays," Nano Lett. 4, 153-158 (2004).
[CrossRef]

Zeng, X. C.

X. C. Zeng, P. M. Hui, D. J. Bergman, and D. Stroud, "Correlation and clustering in the optical properties of composites: a numerical study," Phys. Rev. B 39, 13224-13230 (1989).
[CrossRef]

Int. J. Circuit Theory and Applications (1)

A. I. Csurgay, and W. Porod, "Surface plasmon waves in nanoelectronic circuits," Int. J. Circuit Theory and Applications 32, 339-361 (2004).Q1
[CrossRef]

J. Opt. Soc. Am. B (1)

Nano Lett. (1)

D. A. Genov, A. K. Sarychev, V. M. Shalaev, and A. Wei, "Resonant field enhancement from metal nanoparticle arrays," Nano Lett. 4, 153-158 (2004).
[CrossRef]

Opt. Express (1)

Phys. Rev. B (3)

A. Alù, M. G. Silveirinha, A. Salandrino, and N. Engheta, "Epsilon-near-zero metamaterials and electromagnetic sources: tailoring the radiation phase pattern," Phys. Rev. B 75, 155410 (2007).
[CrossRef]

R. W. Rendell, and D. J. Scalapino, "Surface plasmons confined by microstructures on tunnel junctions," Phys. Rev. B 24, 3276-3294 (1981).
[CrossRef]

X. C. Zeng, P. M. Hui, D. J. Bergman, and D. Stroud, "Correlation and clustering in the optical properties of composites: a numerical study," Phys. Rev. B 39, 13224-13230 (1989).
[CrossRef]

Phys. Rev. B. (2)

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]

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]

Phys. Rev. Lett. (2)

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]

M. G. Silveirinha and N. Engheta, "Tunneling of electromagnetic energy through subwavelength channels and bends using ?-near-zero materials," Phys. Rev. Lett. 97, 157403 (2006).
[CrossRef] [PubMed]

Other (6)

M. G. Silveirinha, A. Alù, J. Li, and N. Engheta, "Nanoinsulators and nanoconnectors for optical nanocircuits," submitted for publication. (Manuscript can be viewed at http://arxiv.org/abs/cond-mat/0703600>).

J. D. Jackson, Classical Electrodynamics (Wiley, New York, USA, 1999).

P. M. Morse and H. Feshbach, Methods of Theoretical Physics, Part I (McGraw-Hill, New York, 1953), p. 663.

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

A. Salandrino, A. Alù, and N. Engheta, "Parallel, series, and intermediate interconnections of optical nanocircuit elements - Part 1: Analytical Solution," submitted to J. Opt. Soc. Am. B. (Manuscript can be viewed at http://arxiv.org/abs/0707.1002>)

A. Alù, A. Salandrino, and N. Engheta, "Parallel, series, and intermediate interconnections of optical nanocircuit elements - Part 2: Nanocircuit and Physical Interpretation," submitted to J. Opt. Soc. Am. B. (Manuscript can be viewed at http://arxiv.org/abs/0707.1003>)

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Figures (4)

Fig. 1.
Fig. 1.

(Color online). A nanoparticle illuminated by a uniform optical electric field E 0 (black arrows) may be viewed in terms of the circuit analogy presented in [1] as a lumped impedance Znano excited by the impressed current generator Iimp and loaded with the fringe capacitance associated with its fringe dipolar fields (red arrows).

Fig. 2.
Fig. 2.

(Color online). Following Fig. 1, a coupled nanocircuit in the optical domain, with optical field coupling between two adjacent nanospheres as two coupled nanoelements.

Fig. 3.
Fig. 3.

(Color online). Potential distribution for a pair of “coupled” ellipsoids (yellow) with a 1 = a 2 = 60 nm, b 1 = b 2 = 50 nm, c 1 = c 2 = 5 nm, d = 2 (c 1 + c 2), ε 1 = ε 2 = -ε 0 and: a) E0 d ; b) E0 d. Lighter grays correspond to higher values of the potential.

Fig. 4.
Fig. 4.

(Color online). A nanocircuit element of permittivity ε over a dielectric planar substrate of permittivity εs , formally equivalent to the coupling problem of two nanocircuit elements symmetrically displaced with respect to the interface, as consistent with the image theory.

Equations (31)

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Z nano = ( iωεπR ) 1
I imp = ( ε ε 0 ) π R 2 E 0 ,
Z fring = ( 2 πR ε 0 ) 1
Z nano = U a b c iωεπab ,
Z fringe = U a b c ε 0 πab 1 L z ( 0 ) L z ( 0 ) ,
I imp = ( ε ε 0 ) πab E 0
C eq , ellipsoid = πab Im ( ε ) U a b c if Re ( ε ) > 0 ,
L eq , ellipsoid = U a b c ω 2 πab Re ( ε ) if Re ( ε ) < 0 ,
G eq , ellipsoid = ωπab Im ( ε ) U a b c ,
C eq , fringe = ε o πab 1 L z ( 0 ) L z ( 0 ) U a b c .
ε = 1 L z ( 0 ) L z ( 0 ) ε 0 ,
E 1 in = 2 ε 0 ε 1 + 2 ε 0 ( E 0 + E 12 )
{ E 1 out = E 0 + 3 r r ( p 1 r r ) p 1 4 π ε 0 r 3 p 1 = 4 π ε 0 ε 1 ε 0 ε 1 + 2 ε 0 R 1 3 ( E 0 + E 12 ) .
E 2 in = 3 ε 0 ε 2 + 2 ε 0 ( E 0 + E 21 )
{ E 2 out = E 0 + 3 r d r d ( p 2 r d r d ) p 2 4 π ε 0 r d 3 p 2 = 4 π ε 0 ε 2 ε 0 ε 2 + ε 0 R 2 3 ( E 0 + E 21 ) .
E 12 E 21 = ( d 3 I ̅ γ 2 ( 3 D ̅ I ̅ ) γ 1 ( 3 D ̅ I ̅ ) d 3 I ̅ ) 1 ( γ 2 ( 3 D ̅ I ̅ ) E 0 γ 1 ( 3 D ̅ I ̅ ) E 0 ) ,
{ E 12 = 2 γ 2 ( d 3 + 2 γ 1 ) d 6 4 γ 1 γ 2 E 0 E 21 = 2 γ 1 ( d 3 + 2 γ 2 ) d 6 4 γ 1 γ 2 E 0 ,
{ E 12 = γ 2 ( γ 1 d 3 ) d 6 γ 1 γ 2 E 0 E 21 = γ 1 ( γ 2 d 3 ) d 6 γ 1 γ 2 E 0 .
I 12 , dip = iωπ E 12 ( ε 1 ε 0 ) R 1 2
I 21 , dip = iωπ E 21 ( ε 2 ε 0 ) R 2 2 ,
E 12 = ( ε 0 ε 1 ) L 1 ( d 2 c 1 2 ) 1 + ( ε 1 ε 0 ) L 1 ( 0 ) [ 1 + ( ε 0 ε 2 ) L 2 ( d 2 c 2 2 ) 1 + ( ε 2 ε 0 ) L 2 ( 0 ) ] 1 ( ε 0 ε 1 ) L 1 ( d 2 c 1 2 ) 1 + ( ε 1 ε 0 ) L 1 ( 0 ) ( ε 0 ε 2 ) L 2 ( d 2 c 2 2 ) 1 + ( ε 2 ε 0 ) L 2 ( 0 ) E 0
E 21 = ( ε 0 ε 2 ) L 2 ( d 2 c 2 2 ) 1 + ( ε 2 ε 0 ) L 2 ( 0 ) [ 1 + ( ε 0 ε 1 ) L 1 ( d 2 c 1 2 ) 1 + ( ε 1 ε 0 ) L 1 ( 0 ) ] 1 ( ε 0 ε 1 ) L 1 ( d 2 c 1 2 ) 1 + ( ε 1 ε 0 ) L 1 ( 0 ) ( ε 0 ε 2 ) L 2 ( d 2 c 2 2 ) 1 + ( ε 2 ε 0 ) L 2 ( 0 ) E 0 ,
E 12 = A 1 ( 1 + A 2 ) 1 A 1 A 2 E 0
E 21 = A 2 ( 1 + A 1 ) 1 A 1 A 2 E 0 ,
I 12 , dip = iωπ E 12 ( ε 1 ε 0 ) a 1 c 1
I 21 , dip = iωπ E 21 ( ε 2 ε 0 ) a 2 c 2 ,
p = α ̅ ( E 0 + E image ) .
E image ( r ) = G ̅ ( r ) p image ,
p = ( ε 2 ε 1 ε 2 + ε 1 ) ( t ̂ t ̂ n ̂ n ̂ ) p = A ̅ p ,
p = { α ̅ part + α ̅ part G ̅ ( 2 d ) [ I ̅ A ̅ α ̅ part G ̅ ( 2 d ) ] 1 A ̅ α ̅ part } E inc
p image = { [ I ̅ A ¯ α ̅ part G ̅ ( 2 d ) ] 1 A ̅ α ̅ part } E inc ,

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