Abstract

In this paper we demonstrate the feasibility of using a plasmonic core-shell particle to function as a wavelength dependent switch for integration into nanoantenna structures. First, a quasistatic analysis is performed and the necessary conditions are derived which allow the particle to operate in either a short- or an open-circuit state. These conditions dictate that the core and the shell permittivity values need to have opposite sign. Consequently, at optical wavelengths where noble metals are modeled as Drude dielectrics, these conditions can be easily realized. As a matter of fact, it is demonstrated that a realistic core-shell particle can exhibit both the short- and open-circuit states, albeit at different wavelengths. Our analysis is extended by examining the same problem beyond the quasistatic limit. For this task we utilize an inhomogeneous spherical transmission line representation of the core-shell particle. The conditions are derived for the particle that yield either an input admittance or impedance equal to zero. It is further demonstrated that these conditions are the short wavelength generalization of their quasistatic counterparts.

© 2013 Optical Society of America

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References

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  1. J. R. Wait, Electromagnetic Waves in Stratified Media (Pergamon, 1962).
  2. W. C. Chew, Waves and Fields in Inhomogeneous Media (IEEE, 1995).
  3. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, 1969).
  4. A. Sihvola and L. V. Lindell, “Transmission line analogy for calculating the effective permittivity of mixtures with spherical multilayer scatterers,” J. Electromagnet. Wave.2(2), 741–756 (1988).
  5. A. Sihvola and L. V. Lindell, “Polarizability and effective permittivity of layered and continuously inhomogeneous dielectric spheres,” J. Electromagnet. Wave.3(1), 37–60 (1989).
    [CrossRef]
  6. J. Li, A. Salandrino, and N. Engheta, “Shaping light beams in the nanometer scale: A Yagi-Uda nanoantenna in the optical domain,” Phys. Rev. B76(24), 245403 (2007).
    [CrossRef]
  7. A. Rashidi and H. Mosallaei, “Array of plasmonic particles enabling optical near-field concentration: A non-linear inverse scattering design approach,” Phys. Rev. B82(3), 035117 (2010).
    [CrossRef]
  8. N. Engheta, A. Salandrino, and A. Alù, “Circuit elements at optical frequencies: nanoinductors, nanocapacitors, and nanoresistors,” Phys. Rev. Lett.95(9), 095504 (2005).
    [CrossRef] [PubMed]
  9. H. Kettunen, H. Wallen, and A. Sihvola, “Electrostatic resonances of a negative-permittivity hemisphere,” J. Appl. Phys.103(9), 094112 (2008).
    [CrossRef]
  10. A. Alù and N. Engheta, “Optical nanoswitch: An engineered plasmonic nanoparticle with extreme parameters and giant anisotropy,” New J. Phys.11(1), 013026 (2009).
    [CrossRef]
  11. A. Alù and N. Engheta, “Optical metamaterials based on optical nanocircuits,” Proc. IEEE99(10), 1669–1681 (2011).
    [CrossRef]
  12. A. Sihvola, “Character of surface plasmons in layered spherical structures,” Prog. Electromagnetics Res.62, 317–331 (2006).
    [CrossRef]
  13. P. B. Johnson and R. W. Christy, “Optical constants of noble metals,” Phys. Rev. B6(12), 4370–4379 (1972).
    [CrossRef]
  14. J. R. Wait, “Electromagnetic scattering from a radially inhomogeneous sphere,” Appl. Sci. Res., Sect. B, Electrophys. Acoust. Opt. Math. Methods10(5-6), 441–450 (1962).
    [CrossRef]
  15. A. L. Aden and M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys.22(10), 1242–1246 (1951).
    [CrossRef]
  16. R. E. Diaz and N. G. Alexopoulos, “An analytic continuation method for the analysis and design of dispersive materials,” IEEE Trans. Antenn. Propag.45(11), 1602–1610 (1997).
    [CrossRef]

2011 (1)

A. Alù and N. Engheta, “Optical metamaterials based on optical nanocircuits,” Proc. IEEE99(10), 1669–1681 (2011).
[CrossRef]

2010 (1)

A. Rashidi and H. Mosallaei, “Array of plasmonic particles enabling optical near-field concentration: A non-linear inverse scattering design approach,” Phys. Rev. B82(3), 035117 (2010).
[CrossRef]

2009 (1)

A. Alù and N. Engheta, “Optical nanoswitch: An engineered plasmonic nanoparticle with extreme parameters and giant anisotropy,” New J. Phys.11(1), 013026 (2009).
[CrossRef]

2008 (1)

H. Kettunen, H. Wallen, and A. Sihvola, “Electrostatic resonances of a negative-permittivity hemisphere,” J. Appl. Phys.103(9), 094112 (2008).
[CrossRef]

2007 (1)

J. Li, A. Salandrino, and N. Engheta, “Shaping light beams in the nanometer scale: A Yagi-Uda nanoantenna in the optical domain,” Phys. Rev. B76(24), 245403 (2007).
[CrossRef]

2006 (1)

A. Sihvola, “Character of surface plasmons in layered spherical structures,” Prog. Electromagnetics Res.62, 317–331 (2006).
[CrossRef]

2005 (1)

N. Engheta, A. Salandrino, and A. Alù, “Circuit elements at optical frequencies: nanoinductors, nanocapacitors, and nanoresistors,” Phys. Rev. Lett.95(9), 095504 (2005).
[CrossRef] [PubMed]

1997 (1)

R. E. Diaz and N. G. Alexopoulos, “An analytic continuation method for the analysis and design of dispersive materials,” IEEE Trans. Antenn. Propag.45(11), 1602–1610 (1997).
[CrossRef]

1989 (1)

A. Sihvola and L. V. Lindell, “Polarizability and effective permittivity of layered and continuously inhomogeneous dielectric spheres,” J. Electromagnet. Wave.3(1), 37–60 (1989).
[CrossRef]

1988 (1)

A. Sihvola and L. V. Lindell, “Transmission line analogy for calculating the effective permittivity of mixtures with spherical multilayer scatterers,” J. Electromagnet. Wave.2(2), 741–756 (1988).

1972 (1)

P. B. Johnson and R. W. Christy, “Optical constants of noble metals,” Phys. Rev. B6(12), 4370–4379 (1972).
[CrossRef]

1962 (1)

J. R. Wait, “Electromagnetic scattering from a radially inhomogeneous sphere,” Appl. Sci. Res., Sect. B, Electrophys. Acoust. Opt. Math. Methods10(5-6), 441–450 (1962).
[CrossRef]

1951 (1)

A. L. Aden and M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys.22(10), 1242–1246 (1951).
[CrossRef]

Aden, A. L.

A. L. Aden and M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys.22(10), 1242–1246 (1951).
[CrossRef]

Alexopoulos, N. G.

R. E. Diaz and N. G. Alexopoulos, “An analytic continuation method for the analysis and design of dispersive materials,” IEEE Trans. Antenn. Propag.45(11), 1602–1610 (1997).
[CrossRef]

Alù, A.

A. Alù and N. Engheta, “Optical metamaterials based on optical nanocircuits,” Proc. IEEE99(10), 1669–1681 (2011).
[CrossRef]

A. Alù and N. Engheta, “Optical nanoswitch: An engineered plasmonic nanoparticle with extreme parameters and giant anisotropy,” New J. Phys.11(1), 013026 (2009).
[CrossRef]

N. Engheta, A. Salandrino, and A. Alù, “Circuit elements at optical frequencies: nanoinductors, nanocapacitors, and nanoresistors,” Phys. Rev. Lett.95(9), 095504 (2005).
[CrossRef] [PubMed]

Christy, R. W.

P. B. Johnson and R. W. Christy, “Optical constants of noble metals,” Phys. Rev. B6(12), 4370–4379 (1972).
[CrossRef]

Diaz, R. E.

R. E. Diaz and N. G. Alexopoulos, “An analytic continuation method for the analysis and design of dispersive materials,” IEEE Trans. Antenn. Propag.45(11), 1602–1610 (1997).
[CrossRef]

Engheta, N.

A. Alù and N. Engheta, “Optical metamaterials based on optical nanocircuits,” Proc. IEEE99(10), 1669–1681 (2011).
[CrossRef]

A. Alù and N. Engheta, “Optical nanoswitch: An engineered plasmonic nanoparticle with extreme parameters and giant anisotropy,” New J. Phys.11(1), 013026 (2009).
[CrossRef]

J. Li, A. Salandrino, and N. Engheta, “Shaping light beams in the nanometer scale: A Yagi-Uda nanoantenna in the optical domain,” Phys. Rev. B76(24), 245403 (2007).
[CrossRef]

N. Engheta, A. Salandrino, and A. Alù, “Circuit elements at optical frequencies: nanoinductors, nanocapacitors, and nanoresistors,” Phys. Rev. Lett.95(9), 095504 (2005).
[CrossRef] [PubMed]

Johnson, P. B.

P. B. Johnson and R. W. Christy, “Optical constants of noble metals,” Phys. Rev. B6(12), 4370–4379 (1972).
[CrossRef]

Kerker, M.

A. L. Aden and M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys.22(10), 1242–1246 (1951).
[CrossRef]

Kettunen, H.

H. Kettunen, H. Wallen, and A. Sihvola, “Electrostatic resonances of a negative-permittivity hemisphere,” J. Appl. Phys.103(9), 094112 (2008).
[CrossRef]

Li, J.

J. Li, A. Salandrino, and N. Engheta, “Shaping light beams in the nanometer scale: A Yagi-Uda nanoantenna in the optical domain,” Phys. Rev. B76(24), 245403 (2007).
[CrossRef]

Lindell, L. V.

A. Sihvola and L. V. Lindell, “Polarizability and effective permittivity of layered and continuously inhomogeneous dielectric spheres,” J. Electromagnet. Wave.3(1), 37–60 (1989).
[CrossRef]

A. Sihvola and L. V. Lindell, “Transmission line analogy for calculating the effective permittivity of mixtures with spherical multilayer scatterers,” J. Electromagnet. Wave.2(2), 741–756 (1988).

Mosallaei, H.

A. Rashidi and H. Mosallaei, “Array of plasmonic particles enabling optical near-field concentration: A non-linear inverse scattering design approach,” Phys. Rev. B82(3), 035117 (2010).
[CrossRef]

Rashidi, A.

A. Rashidi and H. Mosallaei, “Array of plasmonic particles enabling optical near-field concentration: A non-linear inverse scattering design approach,” Phys. Rev. B82(3), 035117 (2010).
[CrossRef]

Salandrino, A.

J. Li, A. Salandrino, and N. Engheta, “Shaping light beams in the nanometer scale: A Yagi-Uda nanoantenna in the optical domain,” Phys. Rev. B76(24), 245403 (2007).
[CrossRef]

N. Engheta, A. Salandrino, and A. Alù, “Circuit elements at optical frequencies: nanoinductors, nanocapacitors, and nanoresistors,” Phys. Rev. Lett.95(9), 095504 (2005).
[CrossRef] [PubMed]

Sihvola, A.

H. Kettunen, H. Wallen, and A. Sihvola, “Electrostatic resonances of a negative-permittivity hemisphere,” J. Appl. Phys.103(9), 094112 (2008).
[CrossRef]

A. Sihvola, “Character of surface plasmons in layered spherical structures,” Prog. Electromagnetics Res.62, 317–331 (2006).
[CrossRef]

A. Sihvola and L. V. Lindell, “Polarizability and effective permittivity of layered and continuously inhomogeneous dielectric spheres,” J. Electromagnet. Wave.3(1), 37–60 (1989).
[CrossRef]

A. Sihvola and L. V. Lindell, “Transmission line analogy for calculating the effective permittivity of mixtures with spherical multilayer scatterers,” J. Electromagnet. Wave.2(2), 741–756 (1988).

Wait, J. R.

J. R. Wait, “Electromagnetic scattering from a radially inhomogeneous sphere,” Appl. Sci. Res., Sect. B, Electrophys. Acoust. Opt. Math. Methods10(5-6), 441–450 (1962).
[CrossRef]

Wallen, H.

H. Kettunen, H. Wallen, and A. Sihvola, “Electrostatic resonances of a negative-permittivity hemisphere,” J. Appl. Phys.103(9), 094112 (2008).
[CrossRef]

Appl. Sci. Res., Sect. B, Electrophys. Acoust. Opt. Math. Methods (1)

J. R. Wait, “Electromagnetic scattering from a radially inhomogeneous sphere,” Appl. Sci. Res., Sect. B, Electrophys. Acoust. Opt. Math. Methods10(5-6), 441–450 (1962).
[CrossRef]

IEEE Trans. Antenn. Propag. (1)

R. E. Diaz and N. G. Alexopoulos, “An analytic continuation method for the analysis and design of dispersive materials,” IEEE Trans. Antenn. Propag.45(11), 1602–1610 (1997).
[CrossRef]

J. Appl. Phys. (2)

A. L. Aden and M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys.22(10), 1242–1246 (1951).
[CrossRef]

H. Kettunen, H. Wallen, and A. Sihvola, “Electrostatic resonances of a negative-permittivity hemisphere,” J. Appl. Phys.103(9), 094112 (2008).
[CrossRef]

J. Electromagnet. Wave. (2)

A. Sihvola and L. V. Lindell, “Transmission line analogy for calculating the effective permittivity of mixtures with spherical multilayer scatterers,” J. Electromagnet. Wave.2(2), 741–756 (1988).

A. Sihvola and L. V. Lindell, “Polarizability and effective permittivity of layered and continuously inhomogeneous dielectric spheres,” J. Electromagnet. Wave.3(1), 37–60 (1989).
[CrossRef]

New J. Phys. (1)

A. Alù and N. Engheta, “Optical nanoswitch: An engineered plasmonic nanoparticle with extreme parameters and giant anisotropy,” New J. Phys.11(1), 013026 (2009).
[CrossRef]

Phys. Rev. B (3)

J. Li, A. Salandrino, and N. Engheta, “Shaping light beams in the nanometer scale: A Yagi-Uda nanoantenna in the optical domain,” Phys. Rev. B76(24), 245403 (2007).
[CrossRef]

A. Rashidi and H. Mosallaei, “Array of plasmonic particles enabling optical near-field concentration: A non-linear inverse scattering design approach,” Phys. Rev. B82(3), 035117 (2010).
[CrossRef]

P. B. Johnson and R. W. Christy, “Optical constants of noble metals,” Phys. Rev. B6(12), 4370–4379 (1972).
[CrossRef]

Phys. Rev. Lett. (1)

N. Engheta, A. Salandrino, and A. Alù, “Circuit elements at optical frequencies: nanoinductors, nanocapacitors, and nanoresistors,” Phys. Rev. Lett.95(9), 095504 (2005).
[CrossRef] [PubMed]

Proc. IEEE (1)

A. Alù and N. Engheta, “Optical metamaterials based on optical nanocircuits,” Proc. IEEE99(10), 1669–1681 (2011).
[CrossRef]

Prog. Electromagnetics Res. (1)

A. Sihvola, “Character of surface plasmons in layered spherical structures,” Prog. Electromagnetics Res.62, 317–331 (2006).
[CrossRef]

Other (3)

J. R. Wait, Electromagnetic Waves in Stratified Media (Pergamon, 1962).

W. C. Chew, Waves and Fields in Inhomogeneous Media (IEEE, 1995).

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, 1969).

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Figures (7)

Fig. 1
Fig. 1

Spherical core-shell particle geometry.

Fig. 2
Fig. 2

Electric field distribution in the space outside the particle under (a) short-circuit conditions and (b) open-circuit conditions. Potential distribution on and around the particle under (c) short-circuit conditions and (d) open-circuit conditions.

Fig. 3
Fig. 3

(a) Core dispersive Drude permittivity along with the necessary permittivity values required to achieve short- and open-circuit conditions. (b) Core-shell effective dielectric properties. (c) Core-shell quasistatic polarizability, real part. (d) Core-shell quasistatic polarizability, imaginary part.

Fig. 4
Fig. 4

(a) TMr reactance of the core-shell particle along with minus the TMr reactance of the fringing field. (b) Extinction efficiency of the core-shell particle. (c) TEr reactance of the core-shell particle along with minus the TEr reactance of the fringing field.

Fig. 5
Fig. 5

Core-shell effective permittivity computed according to Eq. (49) and Eq. (11). (a) Real part. (b) Imaginary part.

Fig. 6
Fig. 6

Comparison between the input impedance of the core-shell particle and an equivalent effective homogeneous dielectric sphere. (a) ZTM resistance. (b) ZTM reactance. (c) ZTE resistance. (d) ZTE reactance.

Fig. 7
Fig. 7

Extinction efficiency comparison: core-shell vs. effective homogenous sphere with permittivity given by Eq. (49).

Equations (51)

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φ 2 = E 0 3( ε 1 +2 ε 2 )3( ε 1 ε 2 ) ( a 1 /r ) 3 ( ε 1 +2 )( ε 1 +2 ε 2 )+2f( ε 1 1 )( ε 1 ε 2 ) rcosθ
φ 2 ( r= a 2 )=0
r φ 2 ( r= a 2 )=0
r ^ ×E( r= a 2 )=0 r ^ × θ ^ 1 a 2 θ φ 2 ( r= a 2 )=0
r ^ E( r= a 2 )=0 r ^ φ 2 ( r= a 2 )=0 r φ 2 ( r= a 2 )=0
ε 1 ε 2 ε 1 +2 ε 2 = 1 f
ε 1 ε 2 ε 1 +2ε 2 = 1 2f
α=4π a 2 3 ε 2 1+f( 2 ε 2 +1 ) α 12 ε 2 +2+2f( ε 2 1 ) α 12
α 12 ε 1 ε 2 ε 1 +2 ε 2
α=4π a 2 3 ε 2 1+2f α 12 1f α 12 1 ε 2 1+2f α 12 1f α 12 +2
ε e = ε 2 1+2f α 12 1f α 12
lim α 12 f ± 1 ε e = ε 2 1+2f( f ± 1 ) 1f( f ± 1 ) ε e ± 
ε e = ε 2 1+2f ( 2f ) 1 1f ( 2f ) 1 ε e =0 
ε 1 ( ω )= ε 2 1+ 2 f 1 1 f
ε 1 ( ω )= ε 2 1 1 f 1+ 1 2f
ε r Ag = ε + f p 2 jvf f 2
Y 2n (e) =j m 2 ψ n ( m 2 ν ) A n χ n ( m 2 v ) ψ n ( m 2 ν ) A n χ n ( m 2 v )
Z 2n (m) = j m 2 ψ n ( m 2 ν ) B n χ n ( m 2 v ) ψ n ( m 2 ν ) B n χ n ( m 2 v )
A n = m 2 ψ n ( m 1 α ) ψ n ( m 2 α ) m 1 ψ n ( m 2 α ) ψ n ( m 1 α ) m 2 ψ n ( m 1 α ) χ n ( m 2 α ) m 1 χ n ( m 2 α ) ψ n ( m 1 α )
B n = m 1 ψ n ( m 1 α ) ψ n ( m 2 α ) m 2 ψ n ( m 2 α ) ψ n ( m 1 α ) m 1 ψ n ( m 1 α ) χ n ( m 2 α ) m 2 χ n ( m 2 α ) ψ n ( m 1 α )
ψ n ( ρ )= πρ 2 J n+1/2 ( ρ )
χ n ( ρ )= πρ 2 H n+1/2 (2) ( ρ )
Y ext,n (e) ( ν )= 1 j χ n ( ν ) χ n ( ν )
Im{ Z 2n (e) + Z ext,n (e) ( ν ) }=0
Z ext,n (m) ( ν )= 1 j χ n ( ν ) χ n ( ν )
Y 1n (e) =j m 1 ψ n ( m 1 α ) ψ n ( m 1 α )
Z 1n ( m ) = j m 1 ψ n ( m 1 α ) ψ n ( m 1 α )
Y ext,n (e) ( m 2 α )= m 2 j χ n ( m 2 α ) χ n ( m 2 α )
Z ext,n (m) ( m 2 α )= 1 j m 2 χ n ( m 2 α ) χ n ( m 2 α )
Y 0,n (e) ( m k 3 r 0 )=jm ψ n ( m k 3 r 0 ) ψ n ( m k 3 r 0 )
Z 0,n (m) ( m k 3 r 0 )= j m ψ n ( m k 3 r 0 ) ψ n ( m k 3 r 0 )
A n (d) = ψ n ( m 2 α ) χ n ( m 2 α ) j m 2 ψ n ( m 2 α ) ψ n ( m 2 α ) +j m 1 ψ n ( m 1 α ) ψ n ( m 1 α ) m 2 j χ n ( m 2 α ) χ n ( m 2 α ) +j m 1 ψ n ( m 1 α ) ψ n ( m 1 α )
A n (u) = ψ n ( m 2 α ) χ n ( m 2 α ) 1 j m 1 ψ n ( m 1 α ) ψ n ( m 1 α ) 1 j m 2 ψ n ( m 2 α ) ψ n ( m 2 α ) 1 j m 1 ψ n ( m 1 α ) ψ n ( m 1 α ) + j m 2 χ n ( m 2 α ) χ n ( m 2 α )
B n (d) = ψ n ( m 2 α ) χ n ( m 2 α ) j m 2 ψ n ( m 2 α ) ψ n ( m 2 α ) j m 1 ψ n ( m 1 α ) ψ n ( m 1 α ) j m 2 χ n ( m 2 α ) χ n ( m 2 α ) j m 1 ψ n ( m 1 α ) ψ n ( m 1 α )
B n (u) = ψ n ( m 2 α ) χ n ( m 2 α ) m 1 j ψ n ( m 1 α ) ψ n ( m 1 α ) m 2 j ψ n ( m 2 α ) ψ n ( m 2 α ) m 1 j ψ n ( m 1 α ) ψ n ( m 1 α ) m 2 j χ n ( m 2 α ) χ n ( m 2 α )
Y 2n (e) =j m 2 ψ n ( m 2 ν ) ψ n ( m 2 ν ) 1 A n (u) χ n ( m 2 v ) ψ n ( m 2 ν ) 1 A n (d) χ n ( m 2 v ) ψ n ( m 2 ν )
Z 2n (m) = j m 2 ψ n ( m 2 ν ) ψ n ( m 2 ν ) 1 B n (u) χ n ( m 2 v ) ψ n ( m 2 ν ) 1 B n (d) χ n ( m 2 v ) ψ n ( m 2 ν )
Ψ( x,y ) χ n ( x ) ψ n ( x ) ψ n ( y ) χ n ( y )
Ω( x,y ) χ n ( x ) ψ n ( x ) ψ n ( y ) χ n ( y )
Y 2n (e) = Y 0,n (e) ( m 2 ν ) 1Ψ( m 2 ν, m 2 α ) 1 Y 1n (e) 1 Y 0,n (e) ( m 2 α ) 1 Y 1n (e) + 1 Y ext,n (e) ( m 2 α ) 1Ω( m 2 ν, m 2 α ) Y 0,n (e) ( m 2 α )+ Y 1n (e) Y ext,n (e) ( m 2 α )+ Y 1n (e)
Z 2n (m) = Z 0,n (m) ( m 2 ν ) 1Ψ( m 2 ν, m 2 α ) 1 Z 1n (m) 1 Z 0,n (m) ( m 2 α ) 1 Z 1n (m) + 1 Z ext,n (m) ( m 2 α ) 1Ω( m 2 ν, m 2 α ) Z 1n (m) Z 0,n (m) ( m 2 α ) Z 1n (m) + Z ext,n (m) ( m 2 α )
Y 1n (e) Y 0,n (e) ( m 2 α ) Y 1n (e) + Y ext,n (e) ( m 2 α ) = 1 Ω( m 2 ν, m 2 α )
Y 1n (e) Y 0,n (e) ( m 2 α ) Y 1n (e) + Y ext,n (e) ( m 2 α ) = Y 0,n (e) ( m 2 α ) Y ext,n (e) ( m 2 α ) 1 Ψ( m 2 ν, m 2 α )
1 Ω( m 2 ν, m 2 α ) = 1 f +O[ ( m 2 α ) 2 ]
Y 0,n (e) ( m 2 α ) Y ext,n (e) ( m 2 α ) 1 Ψ( m 2 ν, m 2 α ) = 1 2f +O[ ( m 2 α ) 2 ]
Y 1n (e) Y 0,n (e) ( m 2 α ) Y 1n (e) + Y ext,n (e) ( m 2 α ) = m 1 ( m 1 α ) m 2 ( m 2 α )+O[ ( m 1 α) 3 + ( m 2 α) 3 ] m 1 ( m 1 α )+2 m 2 ( m 2 α )+O[ ( m 1 α) 3 + ( m 2 α) 3 ]
Y 1n (e) Y 0,n (e) ( m 2 α ) Y 1n (e) + Y ext,n (e) ( m 2 α ) ε 1 ε 2 ε 1 +2 ε 2
Y 2n (e) Y 0 jω a 2 ε 0 2 ε 2 1+2f α 12 1f α 12
ε e = ε + i=1 N 1 L i 1 L i C i +jω R i L i ω 2
Z 0,n (m) ( m k 3 r 0 )= jm k 3 r 0 2m {1+O[ (m k 3 r) 2 ]}
Z 2n (m) Z 0,n (m) ( m 2 ν )

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