Abstract

We present a systematic study of the enhancement of radiative efficiency of light-emitting matter achieved by proximity to metal nanoparticles. Our goal is to ascertain the limits of the attainable enhancement. Two separate arrangements of metal nanoparticles are studied, namely isolated particles and an array of particles. The method of analysis is based on the effective mode volume theory. Using the example of an InGaNGaN quantum-well active region positioned in close proximity to Ag nanospheres, we obtain optimal parameters for the nanoparticles for maximum attainable enhancement. Our results show that while the enhancement due to isolated metal nanoparticles is significant, only modest enhancement can be achieved with an ordered array. We further conclude that a random assembly of isolated particles holds an advantage over the ordered arrays for light-emitting devices of finite area.

© 2008 Optical Society of America

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  1. P. J. Schuck, D. P. Fromm, A. Sundaramurthy, G. S. Kino, W. E. Moerner, “Improving the mismatch between light and nanoscale objects with gold bowtie nanoantennas,” Phys. Rev. Lett. 94, 017402 (2005).
    [CrossRef] [PubMed]
  2. S. Kühn, U. Håkanson, L. Rogobete, V. Sandoghdar, “Enhancement of single-molecule fluorescence using a gold nanoparticle as an optical nanoantenna,” Phys. Rev. Lett. 97, 017402 (2006).
    [CrossRef] [PubMed]
  3. R. Carminati, J.-J. Greffet, C. Henkel, J. M. Vigoureux, “Radiative and nonradiative decay of a single molecule close to a metallic nanoparticle,” Opt. Commun. 261, 368–375 (2006).
    [CrossRef]
  4. L. Rogobete, F. Kaminski, M. Agio, V. Sandoghdar, “Design of plasmonic nanoantennae for enhancing spontaneous emission,” Opt. Lett. 32, 1623–1625 (2007).
    [CrossRef] [PubMed]
  5. M. Thomas, J.-J. Greffet, R. Carminatia, J. R. Arias-Gonzalez, “Single-molecule spontaneous emission close to absorbing nanostructures,” Appl. Phys. Lett. 85, 3863–3865 (2004).
    [CrossRef]
  6. T.-K. Lee, J. L. Birman, “Molecule adsorbed on plane metal surface: coupled system eigenstates,” Phys. Rev. B 22, 5953–5960 (1980).
    [CrossRef]
  7. L. Rogobete, H. Schniepp, V. Sandoghdar, C. Henkel, “Spontaneous emission in nanoscopic dielectric particles,” Opt. Lett. 28, 1736–1738 (2003).
    [CrossRef] [PubMed]
  8. C. Girard, O. J. F. Martin, A. Dereux, “Molecular lifetime changes induced by nanometer scale optical fields,” Phys. Rev. Lett. 75, 3098–3101 (1995).
    [CrossRef] [PubMed]
  9. L. Novotny, “Single molecule fluorescence in inhomogeneous environments,” Appl. Phys. Lett. 69, 3806–3808 (1996).
    [CrossRef]
  10. V. V. Klimov, M. Ducloy, V. S. Letokhov, “Spontaneous emission rate and level shift of an atom inside a dielectric microsphere,” J. Mod. Opt. 43, 549–563 (1996).
    [CrossRef]
  11. A. Rahmani, P. C. Chaumet, F. de Fornel, “Environment-induced modification of spontaneous emission: single-molecule near-field probe,” Phys. Rev. A 63, 023819 (2001).
    [CrossRef]
  12. G. Sun, J. B. Khurgin, R. A. Soref, “Practicable enhancement of spontaneous emission using surface plasmons,” Appl. Phys. Lett. 90, 111107 (2007).
    [CrossRef]
  13. J. B. Khurgin, G. Sun, R. A. Soref, “Enhancement of luminescence efficiency using surface plasmon polaritons—figures of merit,” J. Opt. Soc. Am. B 24, 1968–1980 (2007).
    [CrossRef]
  14. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69, 681 (1946).
    [CrossRef]
  15. S. Maier, “Plasmonic field enhancement and SERS in the effective mode volume picture,” Opt. Express 14, 1957–1964 (2006).
    [CrossRef] [PubMed]
  16. J. D. Jackson, Classical Electrodynamics2nd ed. (John Wiley & Sons, 1962) pp. 150 and 396.

2007 (3)

2006 (3)

S. Maier, “Plasmonic field enhancement and SERS in the effective mode volume picture,” Opt. Express 14, 1957–1964 (2006).
[CrossRef] [PubMed]

S. Kühn, U. Håkanson, L. Rogobete, V. Sandoghdar, “Enhancement of single-molecule fluorescence using a gold nanoparticle as an optical nanoantenna,” Phys. Rev. Lett. 97, 017402 (2006).
[CrossRef] [PubMed]

R. Carminati, J.-J. Greffet, C. Henkel, J. M. Vigoureux, “Radiative and nonradiative decay of a single molecule close to a metallic nanoparticle,” Opt. Commun. 261, 368–375 (2006).
[CrossRef]

2005 (1)

P. J. Schuck, D. P. Fromm, A. Sundaramurthy, G. S. Kino, W. E. Moerner, “Improving the mismatch between light and nanoscale objects with gold bowtie nanoantennas,” Phys. Rev. Lett. 94, 017402 (2005).
[CrossRef] [PubMed]

2004 (1)

M. Thomas, J.-J. Greffet, R. Carminatia, J. R. Arias-Gonzalez, “Single-molecule spontaneous emission close to absorbing nanostructures,” Appl. Phys. Lett. 85, 3863–3865 (2004).
[CrossRef]

2003 (1)

2001 (1)

A. Rahmani, P. C. Chaumet, F. de Fornel, “Environment-induced modification of spontaneous emission: single-molecule near-field probe,” Phys. Rev. A 63, 023819 (2001).
[CrossRef]

1996 (2)

L. Novotny, “Single molecule fluorescence in inhomogeneous environments,” Appl. Phys. Lett. 69, 3806–3808 (1996).
[CrossRef]

V. V. Klimov, M. Ducloy, V. S. Letokhov, “Spontaneous emission rate and level shift of an atom inside a dielectric microsphere,” J. Mod. Opt. 43, 549–563 (1996).
[CrossRef]

1995 (1)

C. Girard, O. J. F. Martin, A. Dereux, “Molecular lifetime changes induced by nanometer scale optical fields,” Phys. Rev. Lett. 75, 3098–3101 (1995).
[CrossRef] [PubMed]

1980 (1)

T.-K. Lee, J. L. Birman, “Molecule adsorbed on plane metal surface: coupled system eigenstates,” Phys. Rev. B 22, 5953–5960 (1980).
[CrossRef]

1946 (1)

M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69, 681 (1946).
[CrossRef]

Agio, M.

Arias-Gonzalez, J. R.

M. Thomas, J.-J. Greffet, R. Carminatia, J. R. Arias-Gonzalez, “Single-molecule spontaneous emission close to absorbing nanostructures,” Appl. Phys. Lett. 85, 3863–3865 (2004).
[CrossRef]

Birman, J. L.

T.-K. Lee, J. L. Birman, “Molecule adsorbed on plane metal surface: coupled system eigenstates,” Phys. Rev. B 22, 5953–5960 (1980).
[CrossRef]

Carminati, R.

R. Carminati, J.-J. Greffet, C. Henkel, J. M. Vigoureux, “Radiative and nonradiative decay of a single molecule close to a metallic nanoparticle,” Opt. Commun. 261, 368–375 (2006).
[CrossRef]

Carminatia, R.

M. Thomas, J.-J. Greffet, R. Carminatia, J. R. Arias-Gonzalez, “Single-molecule spontaneous emission close to absorbing nanostructures,” Appl. Phys. Lett. 85, 3863–3865 (2004).
[CrossRef]

Chaumet, P. C.

A. Rahmani, P. C. Chaumet, F. de Fornel, “Environment-induced modification of spontaneous emission: single-molecule near-field probe,” Phys. Rev. A 63, 023819 (2001).
[CrossRef]

de Fornel, F.

A. Rahmani, P. C. Chaumet, F. de Fornel, “Environment-induced modification of spontaneous emission: single-molecule near-field probe,” Phys. Rev. A 63, 023819 (2001).
[CrossRef]

Dereux, A.

C. Girard, O. J. F. Martin, A. Dereux, “Molecular lifetime changes induced by nanometer scale optical fields,” Phys. Rev. Lett. 75, 3098–3101 (1995).
[CrossRef] [PubMed]

Ducloy, M.

V. V. Klimov, M. Ducloy, V. S. Letokhov, “Spontaneous emission rate and level shift of an atom inside a dielectric microsphere,” J. Mod. Opt. 43, 549–563 (1996).
[CrossRef]

Fromm, D. P.

P. J. Schuck, D. P. Fromm, A. Sundaramurthy, G. S. Kino, W. E. Moerner, “Improving the mismatch between light and nanoscale objects with gold bowtie nanoantennas,” Phys. Rev. Lett. 94, 017402 (2005).
[CrossRef] [PubMed]

Girard, C.

C. Girard, O. J. F. Martin, A. Dereux, “Molecular lifetime changes induced by nanometer scale optical fields,” Phys. Rev. Lett. 75, 3098–3101 (1995).
[CrossRef] [PubMed]

Greffet, J.-J.

R. Carminati, J.-J. Greffet, C. Henkel, J. M. Vigoureux, “Radiative and nonradiative decay of a single molecule close to a metallic nanoparticle,” Opt. Commun. 261, 368–375 (2006).
[CrossRef]

M. Thomas, J.-J. Greffet, R. Carminatia, J. R. Arias-Gonzalez, “Single-molecule spontaneous emission close to absorbing nanostructures,” Appl. Phys. Lett. 85, 3863–3865 (2004).
[CrossRef]

Håkanson, U.

S. Kühn, U. Håkanson, L. Rogobete, V. Sandoghdar, “Enhancement of single-molecule fluorescence using a gold nanoparticle as an optical nanoantenna,” Phys. Rev. Lett. 97, 017402 (2006).
[CrossRef] [PubMed]

Henkel, C.

R. Carminati, J.-J. Greffet, C. Henkel, J. M. Vigoureux, “Radiative and nonradiative decay of a single molecule close to a metallic nanoparticle,” Opt. Commun. 261, 368–375 (2006).
[CrossRef]

L. Rogobete, H. Schniepp, V. Sandoghdar, C. Henkel, “Spontaneous emission in nanoscopic dielectric particles,” Opt. Lett. 28, 1736–1738 (2003).
[CrossRef] [PubMed]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics2nd ed. (John Wiley & Sons, 1962) pp. 150 and 396.

Kaminski, F.

Khurgin, J. B.

J. B. Khurgin, G. Sun, R. A. Soref, “Enhancement of luminescence efficiency using surface plasmon polaritons—figures of merit,” J. Opt. Soc. Am. B 24, 1968–1980 (2007).
[CrossRef]

G. Sun, J. B. Khurgin, R. A. Soref, “Practicable enhancement of spontaneous emission using surface plasmons,” Appl. Phys. Lett. 90, 111107 (2007).
[CrossRef]

Kino, G. S.

P. J. Schuck, D. P. Fromm, A. Sundaramurthy, G. S. Kino, W. E. Moerner, “Improving the mismatch between light and nanoscale objects with gold bowtie nanoantennas,” Phys. Rev. Lett. 94, 017402 (2005).
[CrossRef] [PubMed]

Klimov, V. V.

V. V. Klimov, M. Ducloy, V. S. Letokhov, “Spontaneous emission rate and level shift of an atom inside a dielectric microsphere,” J. Mod. Opt. 43, 549–563 (1996).
[CrossRef]

Kühn, S.

S. Kühn, U. Håkanson, L. Rogobete, V. Sandoghdar, “Enhancement of single-molecule fluorescence using a gold nanoparticle as an optical nanoantenna,” Phys. Rev. Lett. 97, 017402 (2006).
[CrossRef] [PubMed]

Lee, T.-K.

T.-K. Lee, J. L. Birman, “Molecule adsorbed on plane metal surface: coupled system eigenstates,” Phys. Rev. B 22, 5953–5960 (1980).
[CrossRef]

Letokhov, V. S.

V. V. Klimov, M. Ducloy, V. S. Letokhov, “Spontaneous emission rate and level shift of an atom inside a dielectric microsphere,” J. Mod. Opt. 43, 549–563 (1996).
[CrossRef]

Maier, S.

Martin, O. J. F.

C. Girard, O. J. F. Martin, A. Dereux, “Molecular lifetime changes induced by nanometer scale optical fields,” Phys. Rev. Lett. 75, 3098–3101 (1995).
[CrossRef] [PubMed]

Moerner, W. E.

P. J. Schuck, D. P. Fromm, A. Sundaramurthy, G. S. Kino, W. E. Moerner, “Improving the mismatch between light and nanoscale objects with gold bowtie nanoantennas,” Phys. Rev. Lett. 94, 017402 (2005).
[CrossRef] [PubMed]

Novotny, L.

L. Novotny, “Single molecule fluorescence in inhomogeneous environments,” Appl. Phys. Lett. 69, 3806–3808 (1996).
[CrossRef]

Purcell, M.

M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69, 681 (1946).
[CrossRef]

Rahmani, A.

A. Rahmani, P. C. Chaumet, F. de Fornel, “Environment-induced modification of spontaneous emission: single-molecule near-field probe,” Phys. Rev. A 63, 023819 (2001).
[CrossRef]

Rogobete, L.

Sandoghdar, V.

Schniepp, H.

Schuck, P. J.

P. J. Schuck, D. P. Fromm, A. Sundaramurthy, G. S. Kino, W. E. Moerner, “Improving the mismatch between light and nanoscale objects with gold bowtie nanoantennas,” Phys. Rev. Lett. 94, 017402 (2005).
[CrossRef] [PubMed]

Soref, R. A.

G. Sun, J. B. Khurgin, R. A. Soref, “Practicable enhancement of spontaneous emission using surface plasmons,” Appl. Phys. Lett. 90, 111107 (2007).
[CrossRef]

J. B. Khurgin, G. Sun, R. A. Soref, “Enhancement of luminescence efficiency using surface plasmon polaritons—figures of merit,” J. Opt. Soc. Am. B 24, 1968–1980 (2007).
[CrossRef]

Sun, G.

J. B. Khurgin, G. Sun, R. A. Soref, “Enhancement of luminescence efficiency using surface plasmon polaritons—figures of merit,” J. Opt. Soc. Am. B 24, 1968–1980 (2007).
[CrossRef]

G. Sun, J. B. Khurgin, R. A. Soref, “Practicable enhancement of spontaneous emission using surface plasmons,” Appl. Phys. Lett. 90, 111107 (2007).
[CrossRef]

Sundaramurthy, A.

P. J. Schuck, D. P. Fromm, A. Sundaramurthy, G. S. Kino, W. E. Moerner, “Improving the mismatch between light and nanoscale objects with gold bowtie nanoantennas,” Phys. Rev. Lett. 94, 017402 (2005).
[CrossRef] [PubMed]

Thomas, M.

M. Thomas, J.-J. Greffet, R. Carminatia, J. R. Arias-Gonzalez, “Single-molecule spontaneous emission close to absorbing nanostructures,” Appl. Phys. Lett. 85, 3863–3865 (2004).
[CrossRef]

Vigoureux, J. M.

R. Carminati, J.-J. Greffet, C. Henkel, J. M. Vigoureux, “Radiative and nonradiative decay of a single molecule close to a metallic nanoparticle,” Opt. Commun. 261, 368–375 (2006).
[CrossRef]

Appl. Phys. Lett. (3)

M. Thomas, J.-J. Greffet, R. Carminatia, J. R. Arias-Gonzalez, “Single-molecule spontaneous emission close to absorbing nanostructures,” Appl. Phys. Lett. 85, 3863–3865 (2004).
[CrossRef]

L. Novotny, “Single molecule fluorescence in inhomogeneous environments,” Appl. Phys. Lett. 69, 3806–3808 (1996).
[CrossRef]

G. Sun, J. B. Khurgin, R. A. Soref, “Practicable enhancement of spontaneous emission using surface plasmons,” Appl. Phys. Lett. 90, 111107 (2007).
[CrossRef]

J. Mod. Opt. (1)

V. V. Klimov, M. Ducloy, V. S. Letokhov, “Spontaneous emission rate and level shift of an atom inside a dielectric microsphere,” J. Mod. Opt. 43, 549–563 (1996).
[CrossRef]

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

Opt. Commun. (1)

R. Carminati, J.-J. Greffet, C. Henkel, J. M. Vigoureux, “Radiative and nonradiative decay of a single molecule close to a metallic nanoparticle,” Opt. Commun. 261, 368–375 (2006).
[CrossRef]

Opt. Express (1)

Opt. Lett. (2)

Phys. Rev. (1)

M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69, 681 (1946).
[CrossRef]

Phys. Rev. A (1)

A. Rahmani, P. C. Chaumet, F. de Fornel, “Environment-induced modification of spontaneous emission: single-molecule near-field probe,” Phys. Rev. A 63, 023819 (2001).
[CrossRef]

Phys. Rev. B (1)

T.-K. Lee, J. L. Birman, “Molecule adsorbed on plane metal surface: coupled system eigenstates,” Phys. Rev. B 22, 5953–5960 (1980).
[CrossRef]

Phys. Rev. Lett. (3)

C. Girard, O. J. F. Martin, A. Dereux, “Molecular lifetime changes induced by nanometer scale optical fields,” Phys. Rev. Lett. 75, 3098–3101 (1995).
[CrossRef] [PubMed]

P. J. Schuck, D. P. Fromm, A. Sundaramurthy, G. S. Kino, W. E. Moerner, “Improving the mismatch between light and nanoscale objects with gold bowtie nanoantennas,” Phys. Rev. Lett. 94, 017402 (2005).
[CrossRef] [PubMed]

S. Kühn, U. Håkanson, L. Rogobete, V. Sandoghdar, “Enhancement of single-molecule fluorescence using a gold nanoparticle as an optical nanoantenna,” Phys. Rev. Lett. 97, 017402 (2006).
[CrossRef] [PubMed]

Other (1)

J. D. Jackson, Classical Electrodynamics2nd ed. (John Wiley & Sons, 1962) pp. 150 and 396.

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Figures (11)

Fig. 1
Fig. 1

Illustration of the two-step process, with Step 1 being the transfer of energy from emitter into SP modes and Step 2 being the radiative coupling of SP modes into radiation modes. Coupling rates associated with the various processes are shown.

Fig. 2
Fig. 2

Illustration of an isolated Ag nanosphere placed near an active InGaN QW in GaN.

Fig. 3
Fig. 3

Purcell factor of a single metal nanoparticle as a function of sphere radius.

Fig. 4
Fig. 4

Enhancement factor F single due to a single isolated Ag nanosphere as a function of the sphere radius a for a range of the original radiative efficiency of the QW emitter η rad .

Fig. 5
Fig. 5

Optimal enhancement F single , opt by an optimized single isolated Ag nanosphere as a function of the original radiative efficiency η rad of the QW emitter positioned at some distance d below the metal sphere.

Fig. 6
Fig. 6

2-D ordered array of metal nanoparticles placed in the vicinity of the QW active region of a LED.

Fig. 7
Fig. 7

Dispersion relationship of the SP modes of the 2-D array.

Fig. 8
Fig. 8

Radiative ( F P , rad ) , nonradiative ( F P , nrad ) , and total Purcell factors ( F P , rad + F P , nrad ) for a 2-D array as a function of metal sphere radius a with R = 3 a and d = 10 nm .

Fig. 9
Fig. 9

Enhancement due to a 2-D array of Ag spheres that are separated 10 nm from In Ga N Ga N QW emitters with an original radiative efficiency η rad = 0.01 .

Fig. 10
Fig. 10

Optimization of 2-D array of Ag spheres that are separated 5, 10, and 20 nm from In Ga N Ga N QW emitters as a function of original radiative efficiency η rad .

Fig. 11
Fig. 11

Optimal enhancement due to 2-D array of Ag spheres on In Ga N Ga N QW emitter as a function of the original radiative efficiency for several separation values d.

Equations (41)

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E ( r , θ ) = { 3 ϵ D ϵ M + 2 ϵ D E 0 z ̂ r < a E 0 ( ϵ M ϵ D ϵ M + 2 ϵ D ) a 3 r 3 [ 2 cos θ r ̂ sin θ θ ̂ ] r > a } ,
ϵ M = 1 ω p 2 ω 2 + j ω γ ,
ω o = ω p ( 1 + 2 ϵ D ) 1 2 .
E max = 6 ϵ D E o ϵ M + 2 ϵ D .
U = r < a 1 2 ϵ o ( ω ϵ M ) ω E 2 d 3 r + r > a 1 2 ϵ o ϵ D E 2 d 3 r = 1 2 ϵ o ϵ D E max 2 V eff
V eff = 4 3 π a 3 ( 1 + 1 2 ϵ D ) ,
ρ SP = L ( ω ) V eff ( a a + d ) 6 ,
L ( ω ) = Im [ ( ϵ M ( ω ) + 2 ϵ D ) 1 ] Im [ ( ϵ M ( ω ) + 2 ϵ D ) 1 ] d ω .
L ( ω ) = γ d 2 π ( ω ω o ) 2 + γ d 2 4 ,
γ d = γ nrad + γ rad .
( d U d t ) nrad = r < a 1 2 ϵ o Im [ ( ω ϵ M ) ω ] ω E 2 d 3 r = ( 4 3 π a 3 ) { 1 2 ϵ o Im [ ( ω ϵ M ) ω ] ω E 2 } = γ 2 U .
γ nrad = γ 2 ,
p = 4 π ϵ o ϵ M ϵ D 3 ϵ D a 3 ( 3 ϵ D ϵ M + 2 ϵ D E o ) 2 π ϵ o a 3 E max .
P rad = ( d U d t ) rad = ω o 4 n 3 12 π ϵ o c 3 p 2 = ( 2 π a λ D ) 3 ω o 1 + 2 ϵ D U ,
γ rad = ( 2 π a λ D ) 3 ω o 1 + 2 ϵ D .
η pr = γ rad γ nrad + γ rad = Q χ 3 1 + Q χ 3 ,
Q = 2 ω o ( 1 + 2 ϵ D ) γ .
ρ rad = 1 3 π 2 ( 2 π λ D ) 1 ω o .
F p ( ω o ) = ρ SP ρ rad = [ V eff 1 L ( ω o ) ( a a + d ) 6 ] [ 1 3 π 2 ( 2 π λ D ) 3 1 ω o ] 1 = 9 ϵ D Q χ 3 ( 1 + Q χ 3 ) ( χ χ + χ d ) 6 ,
η SP = τ rad 1 + τ rad 1 F p η pr τ nrad 1 + τ rad 1 ( F p + 1 )
η rad = τ rad 1 τ nrad 1 + τ rad 1 .
F single = η SP η rad = 1 + F p η pr 1 + F p η rad ,
F single 1 + F P η pr = 1 + 9 ϵ D Q 2 ( 1 + Q χ 3 ) 2 ( χ χ + χ d ) 6 .
2 p m , n t 2 = ω o 2 p m , n ω o 2 p m 1 , n E out ( R , θ ) E in ω o 2 p m + 1 , n E out ( R , θ ) E in ω o 2 p m , n 1 E out ( R , θ ) E in ω o 2 p m , n + 1 E out ( R , θ ) E in ,
E out ( R , θ ) = E in a 3 R 3 [ 1 + 3 cos 2 θ ] .
E out ( R , θ ) = E in a 3 R 3 ,
2 p m , n t 2 = ω o 2 p m , n ω o 2 a 3 R 3 ( p m 1 , n + p m + 1 , n + p m , n 1 + p m , n + 1 ) .
p m , n = p o e i ( q x m R + q y n R ω q t )
ω q 2 = ω o 2 ( 1 + 2 a 3 R 3 [ cos ( q x R ) + cos ( q y R ) ] ) ,
Δ ω SP 4 ( a R ) 3 ω o ,
ω o = ( 1 + 4 a 3 R 3 ) 1 2 ω o .
g rad = π k D 2 ( 2 π R ) 2 = π R 2 λ D 2 .
η pr , array = γ rad g rad 1 γ rad g rad 1 + γ 2 = Q χ 3 g rad + Q χ 3 .
γ d , array = ( λ D 2 π R 2 ) γ rad + γ 2
L rad ( ω ) = γ d , array 2 π ( ω ω o ) 2 + γ d , array 2 4
F P , rad = g rad 9 π ϵ D γ Q 4 χ 3 L ¯ rad ( ω o ) a 6 [ ( a + d ) 2 + R 2 6 ] 3 ,
d ¯ 2 = ( a + d ) 2 + R 2 6
L ¯ rad ( ω o ) = 0 k D L rad ( 2 π q ) d q 0 k D 2 π q d q .
L nrad ( ω ) = γ 2 π ( ω ω o ) 2 + γ 2 4 .
F P , nrad = ( 1 g rad ) 9 π ϵ D γ Q 4 χ 3 L ¯ nrad ( ω o ) a 6 [ ( a + d ) 2 + R 2 6 ] 3
F array = 1 + F P , rad η pr , array 1 + ( F P , rad + F P , nrad ) η rad .

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