Abstract

A matrix balanced version of the recursive centered T matrix algorithm applicable to systems possessing resonant interparticle couplings is presented. Possible domains of application include systems containing interacting localized plasmon resonances, surface resonances, and photonic jet phenomena. This method is of particular interest when considering modifications to complex systems. The numerical accuracy of this technique is demonstrated in a study of particles with strongly interacting localized plasmon resonances.

© 2008 Optical Society of America

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  1. B. Stout, J.-C. Auger, and J. Lafait, “A transfer matrix approach to local field calculations in multiple scattering problems,” J. Mod. Opt. 49, 2129-2152 (2002).
    [CrossRef]
  2. J.-C. Auger and B. Stout, “A recursive centered T-matrix algorithm to solve the multiple scattering equation: numerical validation,” J. Quant. Spectrosc. Radiat. Transf. 79-80, 533-547 (2003).
    [CrossRef]
  3. A. Doicu and T. Wriedt, Light Scattering by Systems of Particles (Springer, 2006).
    [CrossRef]
  4. L. Tsang, J. A. Kong, and R. T. Shin, Theory of Microwave Remote Sensing (Wiley, 1985).
  5. W. C. Chew, Waves and Fields in Inhomogeneous Media (IEEE Press, 1994).
  6. M. Lax, “Multiple scattering of waves,” Rev. Mod. Phys. 23, 287-310 (1951).
    [CrossRef]
  7. B. Stout, C. Andraud, S. Stout, and J. Lafait, “Absorption in multiple scattering systems of coated spheres,” J. Opt. Soc. Am. A 20, 1050-1059 (2003).
    [CrossRef]
  8. B. Stout, J.-C. Auger, and J. Lafait, “Individual and aggregate scattering matrices and cross sections: conservation laws and reciprocity,” J. Mod. Opt. 48, 2105-2128 (2001).
    [CrossRef]
  9. D. W. Mackowski, “Calculation of total cross sections in multiple-sphere clusters,” J. Opt. Soc. Am. A 11, 2851-2861 (1994).
    [CrossRef]
  10. O. Moine and B. Stout, “Optical force calculations in arbitrary beams by use of the vector addition theorem,” J. Opt. Soc. Am. B 22, 1620-1631 (2005).
    [CrossRef]
  11. M. I. Mishchenko, L. D. Travis, and A. Lacis, Scattering, Absorption and Emission of Light by Small Particles (Cambridge U. Press, 2002).
  12. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, 1983).
  13. D.E.Gray, ed., American Institute of Physics Handbook, 3rd ed. (McGraw-Hill, 1972).
  14. Z. Li, M. Käll, and H. Xu, “Optical forces on interacting plasmonic nanoparticles in a focused Gaussian beam,” Phys. Rev. B 77, 085412 (2008).
    [CrossRef]
  15. S. Bidault, F. J. G. Abajo, and A. Polman, “Plasmon-based nanolenses assembled on a well-defined DNA template,” J. Am. Chem. Soc. 130, 2750-2751 (2008).
    [CrossRef] [PubMed]

2008 (2)

Z. Li, M. Käll, and H. Xu, “Optical forces on interacting plasmonic nanoparticles in a focused Gaussian beam,” Phys. Rev. B 77, 085412 (2008).
[CrossRef]

S. Bidault, F. J. G. Abajo, and A. Polman, “Plasmon-based nanolenses assembled on a well-defined DNA template,” J. Am. Chem. Soc. 130, 2750-2751 (2008).
[CrossRef] [PubMed]

2005 (1)

2003 (2)

J.-C. Auger and B. Stout, “A recursive centered T-matrix algorithm to solve the multiple scattering equation: numerical validation,” J. Quant. Spectrosc. Radiat. Transf. 79-80, 533-547 (2003).
[CrossRef]

B. Stout, C. Andraud, S. Stout, and J. Lafait, “Absorption in multiple scattering systems of coated spheres,” J. Opt. Soc. Am. A 20, 1050-1059 (2003).
[CrossRef]

2002 (1)

B. Stout, J.-C. Auger, and J. Lafait, “A transfer matrix approach to local field calculations in multiple scattering problems,” J. Mod. Opt. 49, 2129-2152 (2002).
[CrossRef]

2001 (1)

B. Stout, J.-C. Auger, and J. Lafait, “Individual and aggregate scattering matrices and cross sections: conservation laws and reciprocity,” J. Mod. Opt. 48, 2105-2128 (2001).
[CrossRef]

1994 (1)

1951 (1)

M. Lax, “Multiple scattering of waves,” Rev. Mod. Phys. 23, 287-310 (1951).
[CrossRef]

Abajo, F. J. G.

S. Bidault, F. J. G. Abajo, and A. Polman, “Plasmon-based nanolenses assembled on a well-defined DNA template,” J. Am. Chem. Soc. 130, 2750-2751 (2008).
[CrossRef] [PubMed]

Andraud, C.

Auger, J.-C.

J.-C. Auger and B. Stout, “A recursive centered T-matrix algorithm to solve the multiple scattering equation: numerical validation,” J. Quant. Spectrosc. Radiat. Transf. 79-80, 533-547 (2003).
[CrossRef]

B. Stout, J.-C. Auger, and J. Lafait, “A transfer matrix approach to local field calculations in multiple scattering problems,” J. Mod. Opt. 49, 2129-2152 (2002).
[CrossRef]

B. Stout, J.-C. Auger, and J. Lafait, “Individual and aggregate scattering matrices and cross sections: conservation laws and reciprocity,” J. Mod. Opt. 48, 2105-2128 (2001).
[CrossRef]

Bidault, S.

S. Bidault, F. J. G. Abajo, and A. Polman, “Plasmon-based nanolenses assembled on a well-defined DNA template,” J. Am. Chem. Soc. 130, 2750-2751 (2008).
[CrossRef] [PubMed]

Bohren, C. F.

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

Chew, W. C.

W. C. Chew, Waves and Fields in Inhomogeneous Media (IEEE Press, 1994).

Doicu, A.

A. Doicu and T. Wriedt, Light Scattering by Systems of Particles (Springer, 2006).
[CrossRef]

Huffman, D. R.

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

Käll, M.

Z. Li, M. Käll, and H. Xu, “Optical forces on interacting plasmonic nanoparticles in a focused Gaussian beam,” Phys. Rev. B 77, 085412 (2008).
[CrossRef]

Kong, J. A.

L. Tsang, J. A. Kong, and R. T. Shin, Theory of Microwave Remote Sensing (Wiley, 1985).

Lacis, A.

M. I. Mishchenko, L. D. Travis, and A. Lacis, Scattering, Absorption and Emission of Light by Small Particles (Cambridge U. Press, 2002).

Lafait, J.

B. Stout, C. Andraud, S. Stout, and J. Lafait, “Absorption in multiple scattering systems of coated spheres,” J. Opt. Soc. Am. A 20, 1050-1059 (2003).
[CrossRef]

B. Stout, J.-C. Auger, and J. Lafait, “A transfer matrix approach to local field calculations in multiple scattering problems,” J. Mod. Opt. 49, 2129-2152 (2002).
[CrossRef]

B. Stout, J.-C. Auger, and J. Lafait, “Individual and aggregate scattering matrices and cross sections: conservation laws and reciprocity,” J. Mod. Opt. 48, 2105-2128 (2001).
[CrossRef]

Lax, M.

M. Lax, “Multiple scattering of waves,” Rev. Mod. Phys. 23, 287-310 (1951).
[CrossRef]

Li, Z.

Z. Li, M. Käll, and H. Xu, “Optical forces on interacting plasmonic nanoparticles in a focused Gaussian beam,” Phys. Rev. B 77, 085412 (2008).
[CrossRef]

Mackowski, D. W.

Mishchenko, M. I.

M. I. Mishchenko, L. D. Travis, and A. Lacis, Scattering, Absorption and Emission of Light by Small Particles (Cambridge U. Press, 2002).

Moine, O.

Polman, A.

S. Bidault, F. J. G. Abajo, and A. Polman, “Plasmon-based nanolenses assembled on a well-defined DNA template,” J. Am. Chem. Soc. 130, 2750-2751 (2008).
[CrossRef] [PubMed]

Shin, R. T.

L. Tsang, J. A. Kong, and R. T. Shin, Theory of Microwave Remote Sensing (Wiley, 1985).

Stout, B.

O. Moine and B. Stout, “Optical force calculations in arbitrary beams by use of the vector addition theorem,” J. Opt. Soc. Am. B 22, 1620-1631 (2005).
[CrossRef]

B. Stout, C. Andraud, S. Stout, and J. Lafait, “Absorption in multiple scattering systems of coated spheres,” J. Opt. Soc. Am. A 20, 1050-1059 (2003).
[CrossRef]

J.-C. Auger and B. Stout, “A recursive centered T-matrix algorithm to solve the multiple scattering equation: numerical validation,” J. Quant. Spectrosc. Radiat. Transf. 79-80, 533-547 (2003).
[CrossRef]

B. Stout, J.-C. Auger, and J. Lafait, “A transfer matrix approach to local field calculations in multiple scattering problems,” J. Mod. Opt. 49, 2129-2152 (2002).
[CrossRef]

B. Stout, J.-C. Auger, and J. Lafait, “Individual and aggregate scattering matrices and cross sections: conservation laws and reciprocity,” J. Mod. Opt. 48, 2105-2128 (2001).
[CrossRef]

Stout, S.

Travis, L. D.

M. I. Mishchenko, L. D. Travis, and A. Lacis, Scattering, Absorption and Emission of Light by Small Particles (Cambridge U. Press, 2002).

Tsang, L.

L. Tsang, J. A. Kong, and R. T. Shin, Theory of Microwave Remote Sensing (Wiley, 1985).

Wriedt, T.

A. Doicu and T. Wriedt, Light Scattering by Systems of Particles (Springer, 2006).
[CrossRef]

Xu, H.

Z. Li, M. Käll, and H. Xu, “Optical forces on interacting plasmonic nanoparticles in a focused Gaussian beam,” Phys. Rev. B 77, 085412 (2008).
[CrossRef]

J. Am. Chem. Soc. (1)

S. Bidault, F. J. G. Abajo, and A. Polman, “Plasmon-based nanolenses assembled on a well-defined DNA template,” J. Am. Chem. Soc. 130, 2750-2751 (2008).
[CrossRef] [PubMed]

J. Mod. Opt. (2)

B. Stout, J.-C. Auger, and J. Lafait, “A transfer matrix approach to local field calculations in multiple scattering problems,” J. Mod. Opt. 49, 2129-2152 (2002).
[CrossRef]

B. Stout, J.-C. Auger, and J. Lafait, “Individual and aggregate scattering matrices and cross sections: conservation laws and reciprocity,” J. Mod. Opt. 48, 2105-2128 (2001).
[CrossRef]

J. Opt. Soc. Am. A (2)

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

J. Quant. Spectrosc. Radiat. Transf. (1)

J.-C. Auger and B. Stout, “A recursive centered T-matrix algorithm to solve the multiple scattering equation: numerical validation,” J. Quant. Spectrosc. Radiat. Transf. 79-80, 533-547 (2003).
[CrossRef]

Phys. Rev. B (1)

Z. Li, M. Käll, and H. Xu, “Optical forces on interacting plasmonic nanoparticles in a focused Gaussian beam,” Phys. Rev. B 77, 085412 (2008).
[CrossRef]

Rev. Mod. Phys. (1)

M. Lax, “Multiple scattering of waves,” Rev. Mod. Phys. 23, 287-310 (1951).
[CrossRef]

Other (6)

A. Doicu and T. Wriedt, Light Scattering by Systems of Particles (Springer, 2006).
[CrossRef]

L. Tsang, J. A. Kong, and R. T. Shin, Theory of Microwave Remote Sensing (Wiley, 1985).

W. C. Chew, Waves and Fields in Inhomogeneous Media (IEEE Press, 1994).

M. I. Mishchenko, L. D. Travis, and A. Lacis, Scattering, Absorption and Emission of Light by Small Particles (Cambridge U. Press, 2002).

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

D.E.Gray, ed., American Institute of Physics Handbook, 3rd ed. (McGraw-Hill, 1972).

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Figures (8)

Fig. 1
Fig. 1

Schematic of a field incident on a collection of scatterers centered on x 1 , x 2 , , x N . The radii of the respective circumscribing spheres are denoted R 1 , R 2 , , R N .

Fig. 2
Fig. 2

Plot of the spherical Bessel to Hankel function ratio ψ n ( k R ) ξ n ( k R ) occurring in the Mie coefficients when k R = 10 .

Fig. 3
Fig. 3

Total cross section efficiencies Q σ ( π R 2 ) for an isolated 50 nm diameter sphere.

Fig. 4
Fig. 4

Electric field intensity E t 2 E inc 2 in an isolated 50 nm diameter sphere ( λ 0 = 365 nm , N Ag = 0.077 + 1.6 i ). (a) 2D (hot) plot of the electric field intensity in a plane perpendicular to the wave vector and containing the origin (horizontal axis lies along the polarization direction). (b) 1D plot of the field intensity along the line in this plane containing the direction of electric field polarization.

Fig. 5
Fig. 5

Dimensionless cross section efficiencies per particle Q = σ ( 2 π R 2 ) and binding force efficiencies for a dimer of 50 nm diameter spheres ( 1 nm separation). In (a) and (b) the polarization is perpendicular to the symmetry axis, and in (c) and (d) it is parallel to the symmetry axis.

Fig. 6
Fig. 6

Logarithmic scale plots of the field intensity for a two-sphere dimer with λ 0 = 467 nm , N Ag = 0.048 + 2.827 i , and incident light polarized along the sphere axis. (a) 2D plot in the plane containing the centers of the spheres and the polarization vector. (b) 1D logarithmic plot along the symmetry axis of the spheres. (c) and (d) are the same as (a) and (b), respectively, but for a five-sphere chain of spheres at its resonance maximum ( λ 0 = 561 nm , N Ag = 0.0564 + 3.685 i ). (cf. Fig. 7).

Fig. 7
Fig. 7

Total cross section and binding efficiencies for a chain of five “touching” silver spheres 50 nm in diameter ( 1 nm separation). In (a) and (b) the polarization is perpendicular to the symmetry axis, (c) and (d), parallel to the symmetry axis.

Fig. 8
Fig. 8

Total extinction and scattering cross section efficiencies per particle in chains of 10 and 20 particles. In (a) the polarization is perpendicular to the symmetry axis, (b), parallel to the symmetry axis.

Tables (2)

Tables Icon

Table 1 Dimensionless Cross Section Efficiencies per Particle as a Function of VSWF Truncation n max

Tables Icon

Table 2 Individual Absorption Efficiencies Q a , j σ a , j ( π R 2 ) in a Five-Sphere Chain a

Equations (41)

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E i ( r ) = E 0 n = 1 m = n n { Rg [ M n m ( k r ) ] a 1 , n , m + Rg [ N n m ( k r ) ] a 2 , n , m } = E 0 q = 1 2 p = 1 Rg [ Ψ q , p ( k r ) ] a q , p E 0 Rg { Ψ t ( k r ) } a ,
n ( p ) = Int p , m ( p ) = p + n ( n + 1 ) .
E t ( r ) = E i ( r ) + j = 1 N E s ( j ) ( r j ) = E 0 Rg [ Ψ t ( k r ) ] a + E 0 j = 1 N Ψ t ( k r j ) f N ( j ) ,
E exc ( j ) ( r j ) E 0 Rg [ Ψ t ( k r j ) ] e N ( j ) E i ( r ) + l = 1 , l j N E s ( l ) ( r l ) = E 0 Rg [ Ψ t ( k r j ) ] [ J ( j , 0 ) a + l = 1 , l j N H ( j , l ) f N ( l ) ] ,
f N ( j ) = T 1 ( j ) e N ( j ) .
e N ( j ) = J ( j , 0 ) a + l = 1 , l j N H ( j , l ) T 1 ( l ) e N ( l ) j = 1 , , N .
f N ( j ) = T 1 ( j ) J ( j , 0 ) a + T 1 ( j ) l = 1 , l j N H ( j , l ) f N ( l ) j = 1 , , N .
f N ( j ) = k = 1 N T N ( j , k ) a ( k ) a ( k ) J ( k , 0 ) a .
[ T 1 ( j ) ] q , p ; q , p = δ q , q δ p , p T 1 ( j , n ( p ) , q ) ,
Φ n ( z ) ψ n ( z ) ψ n ( z ) , Ψ n ( z ) ξ n ( z ) ξ n ( z ) .
T ( j , n , 1 ) = ψ n ( k R j ) ξ n ( k R j ) ( μ j μ ) Φ n ( k R j ) ρ j Φ n ( ρ j k R j ) ρ j Φ n ( ρ j k R j ) ( μ j μ ) Ψ n ( k R j ) ψ n ( k R j ) ξ n ( k R j ) T ¯ ( j , n , 1 ) ,
T ( j , n , 2 ) = ψ n ( k R j ) ξ n ( k R j ) ( μ j μ ) Φ n ( ρ j k R j ) ρ j Φ n ( k R j ) ρ j Ψ n ( k R j ) ( μ j μ ) Φ n ( ρ j k R j ) ψ n ( k R j ) ξ n ( k R j ) T ¯ ( j , n , 2 ) ,
[ f ¯ ( j ) ] q , p ξ n ( p ) ( k R j ) [ f ( j ) ] q , p ,
[ a ¯ ( j ) ] q , p ψ n ( p ) ( k R j ) [ a ( j ) ] q , p .
T ¯ 1 ( j ) [ ξ ( j ) ] T 1 ( j ) [ ψ ( j ) ] 1 , T ¯ N ( j , k ) [ ξ ( j ) ] T ¯ N ( j , k ) [ ψ ( k ) ] 1 .
f ¯ N ( j ) = k = 1 N T ¯ N ( j , k ) a ¯ ( k ) j = 1 , , N .
e N ( N ) = a ( N ) + j , k = 1 N 1 H ( N , j ) T N 1 ( j , k ) a ( k ) + j , k = 1 N 1 H ( N , j ) T N 1 ( j , k ) H ( k , N ) f N ( N ) .
H ¯ ( j , k ) [ ψ ( j ) ] H ( j , k ) [ ξ ( k ) ] 1 , e ¯ N ( j ) [ ψ ( j ) ] e N ( j ) ,
e ¯ N ( N ) = a ¯ ( N ) + j , k = 1 N 1 H ¯ ( N , j ) T ¯ N 1 ( j , k ) a ¯ ( k ) + j , k = 1 N 1 H ¯ ( N , j ) T ¯ N 1 ( j , k ) H ¯ ( k , N ) f ¯ N ( N ) ,
e ¯ N ( j ) = [ T ¯ 1 ( j ) ] 1 f ¯ N ( j ) .
{ [ T ¯ 1 ( N ) ] 1 j , k = 1 N 1 H ¯ ( N , j ) T ¯ N 1 ( j , k ) H ¯ ( k , N ) } f ¯ N ( N ) = a ¯ ( N ) + j , k = 1 N 1 H ¯ ( N , j ) T ¯ N 1 ( j , k ) a ¯ ( k ) .
T ¯ N ( N , N ) = { [ T ¯ 1 ( N ) ] 1 j , k = 1 N 1 H ¯ ( N , j ) T ¯ N 1 ( j , k ) H ¯ ( k , N ) } 1 .
f ¯ N ( N ) = T ¯ N ( N , N ) a ¯ ( N ) + T ¯ N ( N , N ) j , k = 1 N 1 H ¯ ( N , j ) T ¯ N 1 ( j , k ) a ¯ ( k ) = T ¯ N ( N , N ) a ¯ ( N ) + k = 1 N 1 T ¯ N ( N , k ) a ¯ ( k ) = k = 1 N T ¯ N ( N , k ) a ¯ ( k ) ,
T ¯ N ( N , k ) = T ¯ N ( N , N ) j = 1 N 1 H ¯ ( N , j ) T ¯ N 1 ( j , k ) .
f ¯ N ( j ) = k = 1 N 1 T ¯ N 1 ( j , k ) a ¯ ( k ) + k = 1 N 1 T ¯ N 1 ( j , k ) H ¯ ( k , N ) f ¯ N ( N ) = k = 1 N 1 T ¯ N 1 ( j , k ) a ¯ ( k ) + k = 1 N 1 T ¯ N 1 ( j , k ) H ¯ ( k , N ) T ¯ N ( N , N ) a ¯ ( N ) + l = 1 N 1 k = 1 N 1 T ¯ N 1 ( j , l ) H ¯ ( l , N ) T ¯ N ( N , k ) a ¯ ( k ) = T ¯ N ( j , N ) a ¯ ( N ) + k = 1 N 1 T ¯ N ( j , k ) a ¯ ( k ) = k = 1 N T ¯ N ( j , k ) a ¯ ( k ) ,
T ¯ N ( j , N ) = k = 1 N 1 T ¯ N 1 ( j , k ) H ¯ ( k , N ) T ¯ N ( N , N ) .
T ¯ N ( j , k ) = T ¯ N 1 ( j , k ) + l = 1 N 1 T ¯ N 1 ( j , l ) H ¯ ( l , N ) T ¯ N ( N , k ) .
[ T ¯ 1 ( j ) ] 1 f ¯ N ( j ) k = 1 , k j N H ¯ ( j , k ) f ¯ N ( k ) = a ¯ ( j ) j = 1 , , N ,
[ f ¯ N ( 1 ) f ¯ N ( 2 ) f ¯ N ( N ) ] = [ [ T ¯ 1 ( 1 ) ] 1 H ¯ ( 1 , 2 ) H ¯ ( 1 , N ) H ¯ ( 2 , 1 ) [ T ¯ 1 ( 2 ) ] 1 H ¯ ( 2 , N ) H ¯ ( N , 1 ) H ¯ ( N , 2 ) [ T ¯ 1 ( N ) ] 1 ] 1 [ a ¯ ( 1 ) a ¯ ( 2 ) a ¯ ( N ) ] .
[ f ¯ N ( 1 ) f ¯ N ( 2 ) f ¯ N ( N ) ] = [ T ¯ N ( 1 , 1 ) T ¯ N ( 1 , 2 ) T ¯ N ( 1 , N ) T ¯ N ( 2 , 1 ) T ¯ N ( 2 , 2 ) T ¯ N ( 2 , N ) T ¯ N ( N , 1 ) T ¯ N ( N , 2 ) T ¯ N ( N , N ) ] [ a ¯ ( 1 ) a ¯ ( 2 ) a ¯ ( N ) ] ,
T ¯ 1 ( j ) [ ξ ( j ) ] T 1 ( j ) [ ψ ( j ) ] 1 , H ¯ ( j , k ) [ ψ ( j ) ] H ( j , k ) [ ξ ( k ) ] 1 ,
[ ψ ( j ) ] q , q , p , p = δ q , q δ p , p ψ n ( p ) ( k R j )
[ ξ ( j ) ] q , q , p , p = δ q , q δ p , p ξ n ( p ) ( k R j ) ,
σ ext = 1 k 2 Re [ j = 1 N a ( j ) , f N ( j ) ] , σ scat = 1 k 2 j , k = 1 N f N ( j ) , J ( j , k ) f N ( k ) .
σ a ( j ) = 1 k 2 Re { f N ( j ) , e N ( j ) } 1 k 2 f N ( j ) 2 .
F opt = S inc n med c σ opt ,
F b 1 2 ( F 2 F 1 ) r ̂ pos S inc n med c σ b .
Ψ 1 , p ( k r ) M n m ( k r ) h n + ( k r ) X n m ( θ , ϕ ) ,
Ψ 2 , p ( k r ) N n m ( k r ) 1 k r [ n ( n + 1 ) h n + ( k r ) Y n m ( θ , ϕ ) + [ k r h n + ( k r ) ] Z n m ( θ , ϕ ) ] .
Y n m ( θ , ϕ ) r ̂ Y n m ( θ , ϕ ) , Z n m ( θ , ϕ ) r Y n m ( θ , ϕ ) n ( n + 1 ) ,
X n m ( θ , ϕ ) Z n m ( θ , ϕ ) r ̂ ,

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