Abstract

The interaction of light with a metal nanoshell with an off-center core generates multipoles of all orders. We show here that the matrix elements used to compute the multipole expansion coefficients can be derived analytically and, with this result, we can show explicitly how the dipole and quadrupole terms in the expansion are coupled and give rise to a Fano resonance. We also show that the off-center core significantly increases the electric field enhancement at the shell surface compared to the concentric case, which can be exploited for surface-enhanced sensing. The multipole solutions are confirmed with finite-element calculations.

© 2016 Optical Society of America

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References

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  1. C. Wu, A. B. Khanikaev, R. Adato, N. Arju, A. A. Yanik, H. Altug, and G. Shvets, “Fano-resonant asymmetric metamaterials for ultrasensitive spectroscopy and identification of molecular monolayers,” Nat. Mat. 11, 69–75 (2011).
    [Crossref]
  2. A. E. Miroshnichenko, S. Flach, and Y. S. Kivshar, “Fano resonances in nanoscale structures,” Rev. Mod. Phys. 82, 2257–2298 (2010).
    [Crossref]
  3. A. M. Satanin and Y. S. Joe, “Fano interference and resonances in open systems,” Phys. Rev. B 71, 205417 (2005).
    [Crossref]
  4. Y. Wu and P. Nordlander, “Plasmon hybridization in nanoshells with a nonconcentric core,” J. Chem. Phys. 125, 124708 (2006).
    [Crossref]
  5. M. Kerker, ed. Selected Papers on Surface-Enhanced Raman Scattering (SPIE, 1990).
  6. Y. S. Joe, A. M. Satanin, and C. S. Kim, “Classical analogy of Fano resonances,” Phys. Scripta 74, 259–266 (2006).
    [Crossref]
  7. B. Gallinet and O. J. F. Martin, “Ab initio theory of Fano resonances in plasmonic nanostructures and metamaterials,” Phys. Rev. B 83, 235427 (2011).
    [Crossref]
  8. See www.comsol.com for COMSOL Multiphysics.
  9. M. J. Caola, “Solid harmonics and their addition theorems,” J. Phys. A 11, L23–L25 (1978).
    [Crossref]
  10. J. P. Dahl and M. P. Barnett, “Expansion theorems for solid spherical harmonics,” Mol. Phys. 9, 175–178 (1965).
    [Crossref]

2011 (2)

C. Wu, A. B. Khanikaev, R. Adato, N. Arju, A. A. Yanik, H. Altug, and G. Shvets, “Fano-resonant asymmetric metamaterials for ultrasensitive spectroscopy and identification of molecular monolayers,” Nat. Mat. 11, 69–75 (2011).
[Crossref]

B. Gallinet and O. J. F. Martin, “Ab initio theory of Fano resonances in plasmonic nanostructures and metamaterials,” Phys. Rev. B 83, 235427 (2011).
[Crossref]

2010 (1)

A. E. Miroshnichenko, S. Flach, and Y. S. Kivshar, “Fano resonances in nanoscale structures,” Rev. Mod. Phys. 82, 2257–2298 (2010).
[Crossref]

2006 (2)

Y. Wu and P. Nordlander, “Plasmon hybridization in nanoshells with a nonconcentric core,” J. Chem. Phys. 125, 124708 (2006).
[Crossref]

Y. S. Joe, A. M. Satanin, and C. S. Kim, “Classical analogy of Fano resonances,” Phys. Scripta 74, 259–266 (2006).
[Crossref]

2005 (1)

A. M. Satanin and Y. S. Joe, “Fano interference and resonances in open systems,” Phys. Rev. B 71, 205417 (2005).
[Crossref]

1978 (1)

M. J. Caola, “Solid harmonics and their addition theorems,” J. Phys. A 11, L23–L25 (1978).
[Crossref]

1965 (1)

J. P. Dahl and M. P. Barnett, “Expansion theorems for solid spherical harmonics,” Mol. Phys. 9, 175–178 (1965).
[Crossref]

Adato, R.

C. Wu, A. B. Khanikaev, R. Adato, N. Arju, A. A. Yanik, H. Altug, and G. Shvets, “Fano-resonant asymmetric metamaterials for ultrasensitive spectroscopy and identification of molecular monolayers,” Nat. Mat. 11, 69–75 (2011).
[Crossref]

Altug, H.

C. Wu, A. B. Khanikaev, R. Adato, N. Arju, A. A. Yanik, H. Altug, and G. Shvets, “Fano-resonant asymmetric metamaterials for ultrasensitive spectroscopy and identification of molecular monolayers,” Nat. Mat. 11, 69–75 (2011).
[Crossref]

Arju, N.

C. Wu, A. B. Khanikaev, R. Adato, N. Arju, A. A. Yanik, H. Altug, and G. Shvets, “Fano-resonant asymmetric metamaterials for ultrasensitive spectroscopy and identification of molecular monolayers,” Nat. Mat. 11, 69–75 (2011).
[Crossref]

Barnett, M. P.

J. P. Dahl and M. P. Barnett, “Expansion theorems for solid spherical harmonics,” Mol. Phys. 9, 175–178 (1965).
[Crossref]

Caola, M. J.

M. J. Caola, “Solid harmonics and their addition theorems,” J. Phys. A 11, L23–L25 (1978).
[Crossref]

Dahl, J. P.

J. P. Dahl and M. P. Barnett, “Expansion theorems for solid spherical harmonics,” Mol. Phys. 9, 175–178 (1965).
[Crossref]

Flach, S.

A. E. Miroshnichenko, S. Flach, and Y. S. Kivshar, “Fano resonances in nanoscale structures,” Rev. Mod. Phys. 82, 2257–2298 (2010).
[Crossref]

Gallinet, B.

B. Gallinet and O. J. F. Martin, “Ab initio theory of Fano resonances in plasmonic nanostructures and metamaterials,” Phys. Rev. B 83, 235427 (2011).
[Crossref]

Joe, Y. S.

Y. S. Joe, A. M. Satanin, and C. S. Kim, “Classical analogy of Fano resonances,” Phys. Scripta 74, 259–266 (2006).
[Crossref]

A. M. Satanin and Y. S. Joe, “Fano interference and resonances in open systems,” Phys. Rev. B 71, 205417 (2005).
[Crossref]

Khanikaev, A. B.

C. Wu, A. B. Khanikaev, R. Adato, N. Arju, A. A. Yanik, H. Altug, and G. Shvets, “Fano-resonant asymmetric metamaterials for ultrasensitive spectroscopy and identification of molecular monolayers,” Nat. Mat. 11, 69–75 (2011).
[Crossref]

Kim, C. S.

Y. S. Joe, A. M. Satanin, and C. S. Kim, “Classical analogy of Fano resonances,” Phys. Scripta 74, 259–266 (2006).
[Crossref]

Kivshar, Y. S.

A. E. Miroshnichenko, S. Flach, and Y. S. Kivshar, “Fano resonances in nanoscale structures,” Rev. Mod. Phys. 82, 2257–2298 (2010).
[Crossref]

Martin, O. J. F.

B. Gallinet and O. J. F. Martin, “Ab initio theory of Fano resonances in plasmonic nanostructures and metamaterials,” Phys. Rev. B 83, 235427 (2011).
[Crossref]

Miroshnichenko, A. E.

A. E. Miroshnichenko, S. Flach, and Y. S. Kivshar, “Fano resonances in nanoscale structures,” Rev. Mod. Phys. 82, 2257–2298 (2010).
[Crossref]

Nordlander, P.

Y. Wu and P. Nordlander, “Plasmon hybridization in nanoshells with a nonconcentric core,” J. Chem. Phys. 125, 124708 (2006).
[Crossref]

Satanin, A. M.

Y. S. Joe, A. M. Satanin, and C. S. Kim, “Classical analogy of Fano resonances,” Phys. Scripta 74, 259–266 (2006).
[Crossref]

A. M. Satanin and Y. S. Joe, “Fano interference and resonances in open systems,” Phys. Rev. B 71, 205417 (2005).
[Crossref]

Shvets, G.

C. Wu, A. B. Khanikaev, R. Adato, N. Arju, A. A. Yanik, H. Altug, and G. Shvets, “Fano-resonant asymmetric metamaterials for ultrasensitive spectroscopy and identification of molecular monolayers,” Nat. Mat. 11, 69–75 (2011).
[Crossref]

Wu, C.

C. Wu, A. B. Khanikaev, R. Adato, N. Arju, A. A. Yanik, H. Altug, and G. Shvets, “Fano-resonant asymmetric metamaterials for ultrasensitive spectroscopy and identification of molecular monolayers,” Nat. Mat. 11, 69–75 (2011).
[Crossref]

Wu, Y.

Y. Wu and P. Nordlander, “Plasmon hybridization in nanoshells with a nonconcentric core,” J. Chem. Phys. 125, 124708 (2006).
[Crossref]

Yanik, A. A.

C. Wu, A. B. Khanikaev, R. Adato, N. Arju, A. A. Yanik, H. Altug, and G. Shvets, “Fano-resonant asymmetric metamaterials for ultrasensitive spectroscopy and identification of molecular monolayers,” Nat. Mat. 11, 69–75 (2011).
[Crossref]

J. Chem. Phys. (1)

Y. Wu and P. Nordlander, “Plasmon hybridization in nanoshells with a nonconcentric core,” J. Chem. Phys. 125, 124708 (2006).
[Crossref]

J. Phys. A (1)

M. J. Caola, “Solid harmonics and their addition theorems,” J. Phys. A 11, L23–L25 (1978).
[Crossref]

Mol. Phys. (1)

J. P. Dahl and M. P. Barnett, “Expansion theorems for solid spherical harmonics,” Mol. Phys. 9, 175–178 (1965).
[Crossref]

Nat. Mat. (1)

C. Wu, A. B. Khanikaev, R. Adato, N. Arju, A. A. Yanik, H. Altug, and G. Shvets, “Fano-resonant asymmetric metamaterials for ultrasensitive spectroscopy and identification of molecular monolayers,” Nat. Mat. 11, 69–75 (2011).
[Crossref]

Phys. Rev. B (2)

A. M. Satanin and Y. S. Joe, “Fano interference and resonances in open systems,” Phys. Rev. B 71, 205417 (2005).
[Crossref]

B. Gallinet and O. J. F. Martin, “Ab initio theory of Fano resonances in plasmonic nanostructures and metamaterials,” Phys. Rev. B 83, 235427 (2011).
[Crossref]

Phys. Scripta (1)

Y. S. Joe, A. M. Satanin, and C. S. Kim, “Classical analogy of Fano resonances,” Phys. Scripta 74, 259–266 (2006).
[Crossref]

Rev. Mod. Phys. (1)

A. E. Miroshnichenko, S. Flach, and Y. S. Kivshar, “Fano resonances in nanoscale structures,” Rev. Mod. Phys. 82, 2257–2298 (2010).
[Crossref]

Other (2)

M. Kerker, ed. Selected Papers on Surface-Enhanced Raman Scattering (SPIE, 1990).

See www.comsol.com for COMSOL Multiphysics.

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Figures (5)

Fig. 1.
Fig. 1. Core is displaced from the shell center by L .
Fig. 2.
Fig. 2. Five core offsets.
Fig. 3.
Fig. 3. Magnitude of the radiant mode amplitude for five core offsets.
Fig. 4.
Fig. 4. Profiles of the magnitude of the z -component of the electric field through the particle for different core offsets L . Note the change in the vertical scale.
Fig. 5.
Fig. 5. Addition theorem geometry.

Equations (87)

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ψ p ( r 1 , u 1 ) = E 0 r 1 P 1 ( u 1 ) ,
ψ s ( r 1 , u 1 ) = n = 1 a n R 1 n + 1 r 1 n + 1 P n ( u 1 ) ,
ψ 1 ( r 1 , u 1 ) = n = 1 [ b n R 1 n + 1 r 1 n + 1 + c n r 1 n R 1 n ] P n ( u 1 ) ,
ψ 2 ( r 2 , u 2 ) = n = 1 d n r 2 n R 2 n P n ( u 2 ) .
ψ p ( R 1 , u 1 ) + ψ s ( R 1 , u 1 ) = ψ 1 ( R 1 , u 1 ) ,
ε 0 r 1 [ ψ p ( r 1 , u 1 ) + ψ s ( r 1 , u 1 ) ] | r 1 = R 1 = ε 1 r 1 ψ 1 ( r 1 , u 1 ) | r 1 = R 1 ,
ψ 1 ( r 1 , u 1 ) | r 2 = R 2 = ψ 2 ( R 2 , u 2 ) ,
ε 1 r 2 ψ 1 ( r 1 , u 1 ) | r 2 = R 2 = ε 2 r 2 ψ 2 ( r 2 , u 2 ) | r 2 = R 2 .
E 0 P 1 ( u 1 ) + n = 1 a n P n ( u 1 ) = n = 1 ( b n + c n ) P n ( u 1 ) , ε 0 E 0 P 1 ( u 1 ) ε 0 n = 1 ( n + 1 ) a n P n ( u 1 ) = ε 1 n = 1 [ ( n + 1 ) b n + n c n ] P n ( u 1 ) , n = 1 [ b n R 1 n + 1 r 1 n + 1 + c n r 1 n R 1 n ] P n ( u 1 ) | r 2 = R 2 = n = 1 d n P n ( u 2 ) , ε 1 n = 1 r 2 [ b n R 1 n + 1 r 1 n + 1 + c n r 1 n R 1 n ] P n ( u 1 ) | r 2 = R 2 = ε 2 n = 1 n R 2 d n P n ( u 2 ) ,
1 1 P m ( u ) P n ( u ) d u = δ m n w m ,
E 0 δ 1 m + a m = b m + c m ,
ε 0 E 0 δ 1 m ε 0 ( m + 1 ) a m = ε 1 [ ( m + 1 ) b m + m c m ] ,
n = 1 S m n b n + n = 1 T m n c n = d m ,
ε 1 n = 1 U m n b n + ε 1 n = 1 V m n c n = ε 2 m d m ,
S m n = w m R 1 n + 1 1 1 P n ( u 1 ) r 1 n + 1 | r 2 = R 2 P m ( u 2 ) d u 2 ,
T m n = w m R 1 n 1 1 r 1 n P n ( u 1 ) | r 2 = R 2 P m ( u 2 ) d u 2 ,
U m n = w m R 2 R 1 n + 1 1 1 r 2 [ P n ( u 1 ) r 1 n + 1 ] r 2 = R 2 P m ( u 2 ) d u 2 ,
V m n = w m R 2 R 1 n 1 1 r 2 [ r 1 n P n ( u 1 ) ] r 2 = R 2 P m ( u 2 ) d u 2 .
S m n = ( 1 ) m n m ! ( m n ) ! n ! R 1 n + 1 L m n R 2 m + 1 , m n ,
T m n = n ! ( n m ) ! m ! R 2 m L n m R 1 n , n m ,
U m n = ( 1 ) m n m ! ( m n ) ! n ! ( m + 1 ) R 1 n + 1 L m n R 2 m + 1 , m n ,
V m n = n ! ( n m ) ! m ! m R 2 m L n m R 1 n , n m ,
E 0 δ 1 m + a m = b m + c m ,
ε 0 E 0 δ 1 m ε 0 ( m + 1 ) a m = ε 1 [ ( m + 1 ) b m + m c m ] ,
n = 1 m S m n b n + n = m N T m n c n = d m ,
ε 1 ( m + 1 ) n = 1 m S m n b n + ε 1 m n = m N T m n c n = ε 2 m d m ,
3 ε 0 E 0 δ 1 m = ( m + 1 ) ( ε 0 ε 1 ) b m + [ ( m + 1 ) ε 0 + m ε 1 ] c m ,
0 = [ ( m + 1 ) ε 1 + m ε 2 ] n = 1 m S m n b n + m ( ε 2 ε 1 ) n = m T m n c n .
p m = n = 1 N A m n c n , m = 1,2 , , N ,
A m n [ ( m + 1 ) ε 1 + m ε 2 ] [ ( n + 1 ) ε 0 + n ε 1 ( n + 1 ) ( ε 0 ε 1 ) ] S m n + m ( ε 2 ε 1 ) T m n
p m 3 E 0 ε 0 S m 1 ( m + 1 ) ε 1 + m ε 2 2 ( ε 0 ε 1 ) .
3 ε 0 E 0 = 2 ( ε 0 ε 1 ) b 1 + ( 2 ε 0 + ε 1 ) c 1 ,
0 = ( 2 ε 1 + ε 2 ) S 11 b 1 + ( ε 2 ε 1 ) ( T 11 c 1 + T 12 c 2 ) ,
0 = 3 ( ε 0 ε 1 ) b 2 + ( 3 ε 0 + 2 ε 1 ) c 2 ,
0 = ( 3 ε 1 + 2 ε 2 ) ( S 21 b 1 + S 22 b 2 ) + 2 ( ε 2 ε 1 ) T 22 c 2 ,
ε 1 = ε 0 ( 1 ω p 2 ω 2 + i ω γ )
f = M 1 v 1 + P v 2 ,
0 = M 2 v 2 + Q v 1 ,
M 1 ( ω 2 ω 1 2 , ω 2 2 , ω 1 2 t 3 , ω 2 ω 2 2 ) , P L ˜ ( 0 , 0 ω 2 2 t 3 , 0 ) ,
M 2 ( ω 2 ω 3 2 , ω 4 2 ω 3 2 t 5 , ω 2 ω 4 2 ) , Q 2 L ˜ ( 0 , 0 0 , ω 2 ω 4 2 ) .
a 1 = 1 ω 2 [ ω 1 2 c 1 + ( ω 2 ω 2 2 ) b 1 ] = s T v 1 ,
f = M 1 v 1 ,
0 = M 2 v 2 .
D 1 = ( ω 2 ω 1 2 ) ( ω 2 ω 2 2 ) ω 1 2 ω 2 2 t 3 ,
D 2 = ( ω 2 ω 3 2 ) ( ω 2 ω 4 2 ) ω 3 2 ω 4 2 t 5 ,
D 1 = ( ω 2 ω ¯ 1 2 ) ( ω 2 ω ¯ 2 2 ) ,
D 2 = ( ω 2 ω ¯ 3 2 ) ( ω 2 ω ¯ 4 2 ) .
ω ¯ 1 2 = ω 1 2 + ω 1 2 ω 2 2 t 3 ω 1 2 ω 2 2 ,
ω ¯ 2 2 = ω 2 2 ω 1 2 ω 2 2 t 3 ω 1 2 ω 2 2 ,
ω ¯ 3 2 = ω 3 2 + ω 3 2 ω 4 2 t 5 ω 3 2 ω 4 2 ,
ω ¯ 4 2 = ω 4 2 ω 3 2 ω 4 2 t 5 ω 3 2 ω 4 2 .
v 1 c M 1 1 f ,
M 1 1 = 1 D 1 ( ω 2 ω 2 2 , ω 2 2 ω 1 2 t 3 , ω 2 ω 1 2 ) .
a 1 c s T v 1 c = s T M 1 1 f = E 0 ω 1 2 ( ω 2 ω 2 2 ) ( 1 t 3 ) D 1 .
a 1 s = E 0 ω 1 2 ω 2 ω 1 2 ,
v 1 = ( M 1 P M 2 1 Q ) 1 f ,
P M 2 1 Q = 2 ω 2 2 ω 4 2 ( ω 2 ω 4 2 ) t 3 L ˜ 2 D 2 ( 0 , 0 0 , 1 ) ,
α 2 ω 2 2 ω 4 2 ( ω 2 ω 4 2 ) t 3 .
M 1 P M 2 1 Q = ( ω 2 ω 1 2 , ω 2 2 ω 1 2 t 3 , ω 2 ω 2 2 + α L ˜ 2 / D 2 ) .
( M 1 P M 2 1 Q ) 1 = D 1 D 2 M 1 1 + α L ˜ 2 F D 1 D 2 + α ( ω 2 ω 2 2 ) L ˜ 2 ,
F ( 1 , 0 0 , 0 )
v 1 = ( M 1 P M 2 1 Q ) 1 f = D 1 D 2 M 1 1 f + α L ˜ 2 F f D 1 D 2 + α ( ω 2 ω 2 2 ) L ˜ 2 .
a 1 = D 1 D 2 a 1 c + α L ˜ 2 s T F f D 1 D 2 + α ( ω 2 ω 2 2 ) L ˜ 2 .
a 1 = D 1 D 2 a 1 c α E 0 ω 1 2 L ˜ 2 D 1 D 2 + α ( ω 2 ω 2 2 ) L ˜ 2 .
D D 1 D 2 + α ( ω 2 ω 2 2 ) L ˜ 2 = ( ω 2 ω ¯ 1 2 ) ( ω 2 ω ¯ 2 2 ) ( ω 2 ω ¯ 3 2 ) ( ω 2 ω ¯ 4 2 ) + α ( ω 2 ω 2 2 ) L ˜ 2 ,
a 1 a 1 c = ω 2 ω ¯ 4 2 α E 0 ω 1 2 L ˜ 2 / ( a 1 c C ) + i ω ¯ 4 2 γ ω 2 ω ¯ 4 2 + α ( ω ¯ 4 2 ω 2 2 ) L ˜ 2 / C + i ω ¯ 4 2 γ .
| a 1 a 1 c | 2 = ( κ q ) 2 + 1 κ 2 + 1 ,
( ω 2 ω 1 2 ) a 1 + ω 1 2 a 2 = f , [ ω 2 ( 1 + ε 2 ) ω 2 2 ] a 2 + ω 2 2 a 1 + ε 2 ω 2 2 a 3 = 0 , [ ω 2 ( 1 + ε 2 ) ω 3 2 ] a 3 + ε 2 ω 3 2 a 2 + ω 3 2 a 4 = 0 , ( ω 2 ω 4 2 ) a 4 + ω 4 2 a 3 = 0 ,
D 1 ( ω 2 ω 1 2 ) [ ω 2 ( 1 + ε 2 ) ω 2 2 ] ω 1 2 ω 2 2 ,
D 2 ( ω 2 ω 4 2 ) [ ω 2 ( 1 + ε 2 ) ω 3 2 ] ω 3 2 ω 4 2 ,
a 1 = f D [ ( ω 2 ( 1 + ε 2 ) ω 2 2 ) D 2 ε 4 ω 2 2 ω 3 2 ( ω 2 ω 4 2 ) ] ,
D D 1 D 2 ε 4 ω 2 2 ω 3 2 ( ω 2 ω 1 2 ) ( ω 2 ω 4 2 )
a 1 0 = f [ ω 2 ( 1 + ε 2 ) ω 2 2 ] D 1 .
a 1 = D 1 D 2 a 1 0 + ε 4 f ω 2 2 ω 3 2 ( ω 2 ω 4 2 ) D 1 D 2 ε 4 ω 2 2 ω 3 2 ( ω 2 ω 1 2 ) ( ω 2 ω 4 2 ) ,
R n ( r ) = r n P n ( u ) ,
I n ( r ) = 1 r n + 1 P n ( u ) ,
R n ( r + r ) = k = 0 n n ! k ! ( n k ) ! R k ( r ) R n k ( r ) ,
I n ( r + r ) = k = 0 ( 1 ) k ( n + k ) ! k ! n ! R k ( r ) I n + k ( r ) .
R n ( r 2 + R ) = R n ( r 1 ) = k = 0 n n ! k ! ( n k ) ! R k ( r 2 ) R n k ( R )
r 1 n P n ( u 1 ) = k = 0 n n ! k ! ( n k ) ! r 2 k P k ( u 2 ) L n k .
I n ( r 2 + R ) = I n ( r 1 ) = k = 0 ( 1 ) k ( n + k ) ! k ! n ! R k ( R ) I n + k ( r 2 )
1 r 1 n + 1 P n ( u 1 ) = k = 0 ( 1 ) k ( n + k ) ! k ! n ! L k 1 r 2 n + k + 1 P n + k ( u 2 ) .
ψ ( r ) = 1 4 π S [ g ( r | r ) ϕ ( r ) ψ ( r ) g ( r | r ) ] · n ^ d S ,
g ( r 1 , 1 | R 1 , u 1 ) = 1 r 1 2 + R 1 2 2 r 1 R 1 u 1 .
g ( r 2 , 1 | R 2 , u 2 ) = 1 r 2 2 + R 2 2 2 r 2 R 2 u 2 .
1 1 P n ( u ) d u r 2 + R 2 2 r R u = { ( 2 2 n + 1 ) r n R n + 1 , r < R , ( 2 2 n + 1 ) R n r n + 1 , R < r ,
ψ ( r 1 , 1 ) = E 0 r 1 3 ( 2 + ε 0 ε 1 ) + ( 1 ε 0 ε 1 ) n = 1 N a n ( n + 1 2 n + 1 ) r 1 n R 1 n + ( 1 ε 2 ε 1 ) n = 1 N n d n 2 n + 1 R 2 n + 1 r 2 n + 1 ,

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